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MATHEMATICAL  MONOGRAPHS. 

EDITED    BY 

Mansfield  Merriman  and  Robert  S.  Woodward. 
Octavo,  Cloth,  $1.00  each. 


No.  1.    HISTORY  OF  MODERN  MATHEMATICS. 

By  David  Eugene  Smith. 
No.  2.    SYNTHETIC  PROJECTIVE  QEOMETRV. 
By  George  1?ruce  Halsthd. 
No,  3.    DETERMINANTS. 
By  Laenas  GiFFOKD  Weld. 
VN0.4.    HYPERBOLIC   FUNCTIONS. 
By  James  McMahon. 
V  No.  5.    HARMONIC  FUNCTIONS. 
By  William  E.  Byerly. 
No.  6.    ORASSMANN'S  SPACE  ANALYSIS. 
By  Edward  W.  Hydh. 
No.  7.    PROBABILITY    AND  THEORY    OF    ERRORS. 
By  Robert  S.  Woodward. 
No.  8k.   VECTOR  ANALYSIS  AND  QUATERNIONS. 
By  Alexander  Macfarlanr. 
VNo.  9.    DIFFERENTIAL  EQUATIONS. 
By  William  Woolsev  Johnson. 
/No.  10.    THE  SOLUTION  OF  EQUATIONS. 
By  Mansfield  Merriman. 
No,  11.    FUNCTIONS  OF  A  COMPLEX  VARIABLE. 
By  Thomas  S.  Fiskb. 

PUBLISHED   BY 

JOHN  WILEY  &  SONS,   NEW  YORK. 
CHAPMAN  &  HALL.  Limited,  LONDON. 


MATHEMATICAL    MONOGRAPHS. 

EDITED    BY 

MANSFIELD   MERRIMAN  and  ROBERT   S.  WOODWARD. 


No.  9. 


DIFFERENTIAL   EQUATIONS. 


BY 

W.    WOOLSEY    JOHNSON, 

Professor   of  Mathematics  [n  the  United  States  Naval  Academy. 


FOURTH    EDITION. 
FIRST   THOUSAND. 


NEW  YORK: 

JOHN   WILEY   &    SONS. 

London:    CHAPMAN  &  HALL,    Limited. 

1906. 


(^^'x^. 


Copyright,  1896, 

BY 

MANSFIELD   MERRIMAN  and  ROBERT  S.  WOODWARD 

UNDER  THE   TiTLK 

HIGHER    MATHEMATICS. 


First  Edition,  September,  1896. 
Second  Edition,  January,  1898. 
Third  Edition,  August,  1900. 
Fourth  Edition,  January,  1906. 


ROBERT  DRimMOND,   PRINTER,   NEW   TORK. 


EDITORS'  PREFACE. 


The  volume  called  Higher  Mathematics,  the  first  edition 
of  which  was  published  in  1896,  contained  eleven  chapters  by 
eleven  authors,  each  chapter  being  independent  of  the  others, 
but  all  supposing  the  reader  to  have  at  least  a  mathematical 
training  equivalent  to  that  given  in  classical  and  engineering 
colleges.  The  publication  of  that  volume  is  now  discontinued 
and  the  chapters  are  issued  in  separate  form.  In  these  reissues 
it  will  generally  be  found  that  the  monographs  are  enlarged 
by  additional  articles  or  appendices  which  either  amplify  the 
former  presentation  or  record  recent  advances.  This  plan  of 
publication  has  been  arranged  in  order  to  meet  the  demand  of 
teachers  and  the  convenience  of  classes,  but  it  is  also  thought 
that  it  may  prove  advantageous  to  readers  in  special  lines  of 
mathematical  Hterature. 

It  is  the  intention  of  the  publishers  and  editors  to  add  other 
monographs  to  the  series  from  time  to  time,  if  the  call  for  the 
same  seems  to  warrant  it.  Among  the  topics  which  are  under 
consideration  are  those  of  elHptic  functions,  the  theory  of  num- 
bep,  the  group  theory,  the  calculus  of  variations,  and  non- 
EucHdean  geometry;  possibly  also  monographs  on  branches  of 
astronomy,  mechanics,  and  mathematical  physics  may  be  included. 
It  is  the  hope  of  the  editors  that  this  form  of  pubHcation  may 
tend  to  promote  mathematical  study  and  research  over  a  wider 
field  than  that  which  the  former  volume  has  occupied. 

December,  1905.  "i 


236368 


AUTHOR'S  PREFACE. 


It  is  customary  to  divide  the  Infinitesimal  Calculus,  or  Calcu- 
lus of  Continuous  Functions,  into  three  parts,  under  the  heads 
Differential  Calculus,  Integral  Calculus,  and  Differential  Equa- 
tions. The  first  corresponds,  in  the  language  of  Newton,  to  the 
"direct  method  of  tangents, "  the  other  two  to  the  "inverse  method 
of  tangents";  while  the  questions  which  come  under  this  last 
head  he  further  -divided  into  those  involving  the  two  fluxions  and 
one  fluent,  and  those  involving  the  fluxions  and  both  fluents. 

On  account  of  the  inverse  character  which  thus  attaches  to 
the  present  subject,  the  differential  equation  must  necessarily 
at  first  be  viewed  in  connection  with  a  "primitive,"  from  which 
it  might  have  been  obtained  by  the  direct  process,  and  the  solu- 
tion consists  in  the  discovery,  by  tentative  and  more  or  less  arti- 
ficial methods,  of  such  a  primitive,  when  it  exists;  that  is  to 
say,  when  it  is  expressible  in  the  elementary  functions  which 
constitute  the  original  field  with  which  the  Differential  Calci^Jus 
has  to  do. 

It  is  the  nature  of  an  inverse  process  to  enlarge  the  field  of 
its  operations,  and  the  present  is  no  exception;  but  the  adequate 
handling  of  the  new  functions  with  which  the  field  is  thus  enlarged 
requires  the  introduction  of  the  complex  variable,  and  is  beyond 
the  scope  of  a  work  of  this  size. 

But  the  theory  of  the  nature  and  meaning  of  a  differential 
equation  between  real  variables  possesses  a  great  deal  of  interest. 
To  this  part  of  the  subject  I  have  endeavored  to  give  a  full  treat- 
ment by  means  of  extensive  use  of  graphic  representations  in 


.      AUTHOR  S   PREFACE.  V 

rectangular  coordinates.  If  we  ask  what  it  is  that  satisfies  an 
ordinary  differential  equation  of  the  first  order,  the  answer  must 
be  certain  sets  of  simultaneous  values  of  x,  y,  and  p.  The  geo- 
metrical representation  of  such  a  set  is  a  point  in  a  plane  asso- 
ciated with  a  direction,  so  to  speak,  an  infinitesimal  stroke,  and 
the  "solution"  consists  of  the  grouping  together  of  these  strokes 
into  curves  of  which  they  form  elements.  The  treatment  of 
singular  solutions,  following  Cayley,  and  a  comparison  with  the 
methods  previously  in  use,  illustrates  the  great  utility  of  this  point 
of  view. 

Again,  in  partial  differential  equations,  the  set  of  simultaneous 
values  of  x,  y,  z,  p,  and  q  which  satisfies  an  equation  of  the  first 
order  is  represented  by  a  point  in  space  associated  with  the  direc- 
tion of  a  plane,  so  to  speak  by  a  flake,  and  the  mode  in  which 
these  coalesce  so  as  to  form  linear  surface  elements  and  con- 
tinuous surfaces  throws  light  upon  the  nature  of  general  and 
complete  integrals  and  of  the  characteristics. 

The  expeditious  symbolic  methods  of  integration  applicable 
to  some  forms  of  linear  equations,  and  the  subject  of  development 
of  integrals  in  convergent  series,  have  been  treated  as  fully  as  space 
would  allow. 

Examples  selected  to  illustrate  the  principles  developed  in 
each  section  will  be  found  at  its  close,  and  a  full  index  of  subjects 
at  the  end  of  the  volume. 

W.  W.  J. 

Annapolis,  Md.,  December,  1905. 


CONTENTS. 


Art.  I.  Equations  of  First  Order  and  Degree Page  i 

2.  Geometrical  Representation 3 

3.  Primitive  of  a  Differential  Equation 5 

4.  Exact  Differential  Equations 6 

5.  Homogeneous  Equations 9 

6.  The  Linear  Equation 10 

7.  First  Order  and  Second  Degree 12 

8.  Singular  Solutions 15 

9.  Singular  Solution  from  the  Complete  Integral      .     .     ,     «  18 

10.  Solution  by  Differentiation 20 

11.  Geometric  Applications,  Trajectories 23 

12.  Simultaneous  Differential  Equations 25 

13.  Equations  of  the  Second  Order 28 

14.  The  Two  First  Integrals 31 

15.  Linear  Equations 34 

16.  Linear  Equations  with  Constant  Coefficients 36 

17.  Homogeneous  Linear  Equations 40 

18.  Solutions  in  Infinite  Series 42 

19.  Systems  of  Differential  Equations 47 

20.  First  Order  and  Degree  with  Three  Variables    ....  50 

21.  Partial  Differential  Equations  of  First  Order  and  Degree  53 

22.  Complete  and  General  Integrals 57 

23.  Complete  Integral  for  Special  Forms 60 

24.  Partial  Equations  of  Second  Order 63 

25.  Linear  Partial  Differential  Equations     .......  66 

Index 72 


DIFFERENTIAL    EQUATIONS. 


Art.  1.    Equations  of  First  Order  and  Degree. 

In  the  Integral  Calculus,  supposing  j/  to  denote  an  unknown 
function  of  the  independent  variable  x,  the  derivative  of  j  with 
respect  to  x  is  given  in  the  form  of  a  function  of  x^  and  it  is 
required  to  find  the  value  of  j  as  a  function  of  x.  In  other 
words,  given  an  equation  of  the  form 

of  which  the  general  solution  is  written  in  the  form 

y  =  f/i^y^.  (2> 

it  is  the  object  of  the  Integral  Calculus  to  reduce  the  expres- 
sion in  the  second  member  of  equation  (2)  to  the  form  of  a 
known  function  of  x.  When  such  reduction  is  not  possible, 
the  equation  serves  to  define  a  new  function  of  x. 

In  the  extension  of  the  processes  of  integration  of  which 

the  following  pages  give  a  sketch  the  given  expression  for  the 

derivative  may  involve  not  only  x,  but  the  unknown  function 

y ;  or,  to  write  the  equation  in  a  form  analogous  to  equation 

(i),  it  may  be 

Mdx  +  Ndj^  =  o,  (3) 

in  which  J/ and  iVare  functions  of  x  and^;^.  This  equation  is 
in  fact  the  general  form  of  the  differential  equation  of  the  first 
order  and  degree ;  either  variable  being  taken  as  the  independ- 
ent variable,  it  gives  the  first  derivative  of  the  other  variable 


JJ  DIFFERENTIAL   EQUATIONS. 

in  terms  of  x  and  y.  So  also  the  solution  is  not  necessarily  an 
expression  of  either  variable  as  a  function  of  the  other,  but  is 
generally  a  relation  between  x  and  y  which  makes  either  an 
impHcit  function  of  the  other. 

When  we  recognize  the  left  member  of  equation  (3)  as  an 
"exact  differential,"  that  is,  the  differential  of  some  function  of 
X  and  /,  the  solution  is  obvious.  For  example,  given  the  equa- 
tion 

;r^+j>/^^  =  0,  (4) 

the  solution  xy  =  C,  (5) 

where  C  is  an  arbitrary  constant,  is  obtained  by  "  direct  inte- 
gration." When  a  particular  value  is  attributed  to  C,  the  result 
is  a  "  particular  integral ;  "  thus^  =  ;r"^  is  a  particular  integral 
of  equation  (4),  while  the  more  general  relation  expressed  by 
equation  (5)  is  known  as  the  "  complete  integral." 

In  general,  the  given  expression  Mdx  +  Ndy  is  not  an  ex- 
act differential,  and  it  is  necessary  to  find  some  less  direct 
method  of  solution. 

The  most  obvious  method  of  solving  a  differential  equation 
of  the  first  order  and  degree  is,  when  practicable,  to  "  separate 
the  variables,"  so  that  the  coefficient  of  dx  shall  contain  x 
only,  and  that  of  dy,  y  only.     For  example,  given  the  equation 

(I  - y)dx  +  (I  +  x)dy  =  o,  (6) 

the  variables   are   separated    by  dividing  by   {\ -\-  x){\  —  y). 

Tu                                    dx           dy 
Thus  — ; ^—  =0. 

I  -|-;r;  ^    I  —  y 

Each  term  is  now  directly  integrable,  and  hence 
log  (I +^)  -  log  (i -^)  =  r. 

The  solution  here  presents  itself  in  a  transcendental  form, 
but  it  is  readily  reduced  to  an  algebraic  form.  For,  taking  the 
exponential  of  each  member,  we  find 

ii^  =  ^  =  C,   whence    \ -^  x  =z  C{i  -  y\  (7) 

where  C  is  put  for  the  constant  ^. 


GEOMETRICAL    REPRESENTATION.  3 

To  verify  the  result  in  this  form  we  notice  that  differentia- 
tion gives  dx  =^  —  Cdy,  and  substituting  in  equation  (6)  we  find 

which  is  true  by  equation  (7). 

Prob.  I.  Solve  the  equation  dy  -\- y  tan  x  dx=^  o^ 

An^.  y=-C  cos  x.  y' 

-"Prob.  2.  Solve  ^  +  b'^f  =  a\ 
dx  ^ 


Prob.  3.  Solve  ^  =  =^^4-" 
dx        X   -\-  \ 


Prob.  4.  Helmholtz's  equation  for  the  strength  of  an   electric 
current  C  at  the  time  /  is 

r-  —  -  ^^ 

R       R   dt' 
where  E,  R,  and  L  are  given  constants.     Find  the  value  of  C,  de- 
termining the  constant  of  integration  by  the  condition  that  its  initial 
value  shall  be  zero. 


Art.  2.    Geometrical  Representation. 

The  meaning  of  a  differential  equation  may  be  graphically 
illustrated  by  supposing  simultaneous  values  of  x  and  y  to  be 
the  rectangular  coordinates  of  a  variable  point.  It  is  conven- 
ient to  put/  for  the  value  of  the  ratio  dy  :  dx.  Then  P  being 
the  moving  point  {x,  y)  and  0  denoting  the  inclination  of  its 
path  to  the  axis  of  x,  we  have 

dy 

The  given  differential  equation  of  the  first  order  is  a  relation 
between />,  x,  a-nd  y,  and,  being  of  the  first  degree  with  respect 
to/,  determines  in  general  a  single  value  of/  for  any  assumed 
values  of  x  and  y.  Suppose  in  the  first  place  that,  in  addition 
to  the  differential  equation,  we  were  given  one  pair  of  simul- 
taneous values  of  x  andj^,  that  is,  one  position  of  the  point  P. 
Now  let  P  start  from  this  fixed  initial  point  and  begin  to  move 
in  either  direction  along  the  straight  line  whose   inclination 


DIFFERENTIAL   EQUATIONS. 


is  determined  by  the  value  of  /  corresponding  to  the  initial 
values  of  x  and  y.  We  thus  have  a  moving  point  satisfying 
the  given  differential  equation.  As  the  point  P  moves  the 
values  of  x  and  y  vary,  and  we  must  suppose  the  direction  of 
its  motion  to  vary  in  such  a  way  that  the  simultaneous  values 
of  x,y^  and/  continue  to  satisfy  the  differential  equation.  In 
that  case,  the  path  of  the  moving  point  is  said  to  satisfy  the 
differential  equation.  The  point  P  may  return  to  its  initial 
position,  thus  describing  a  closed  curve,  or  it  may  pass  to  infin- 
ity in  each  direction  from  the  initial  point  describing  an  infinite 
branch  of  a  curve.*  The  ordinary  cartesian  equation  of  the 
path  of  P  is  a  particular  integral  of  the  differential  equation. 

If  no  pair  of  associated  vahies  of  x  and  y  be  known,  P  may 
be  assumed  to  start  from  any  initial  point,  so  that  there  is  an 
unlimited  number  of  curves  representing  particular  integrals 
of  the  equation.  These  form  a  "system  of  curves,"  and  the 
complete  integral  is  the  equation  of  the  system  in  the  usual 
form  of  a  relation  between  x^y,  and  an  arbitrary  "  parameter." 
This  parameter  is  of  course  the  constant  of  integration.  It  is 
constant  for  any  one  curve  of  the  system,  and  different  values 
of  it  determine  different  members  of  the  system  of  curves,  or 
different  particular  integrals. 

As  an  illustration,  let  us  take  equation  (4)  of  Art.  I,  which 

may  be  written 

-  -^  =  _  Z 

dx  '  x' 

Denoting  by  B  the  inclination  to 

the  axis  of  x  of  the  line  joining  P 

with   the   origin,  the  equation   is 

equivalent  to  tan  0  =  —  tan  6,  and 

therefore  expresses  that  P  moves 

in  a  direction  inclined  equally  with 

OP  to  either  axis,  but  on  the  other 

*  When  the  form  of  the  functions  M  and  JV  is  unrestricted,  there  is  no 
reason  why  either  of  these  cases  should  exist,  but  they  commonly  occur  among 
such  differential  equations  as  admit  of  solution. 


PRIMITIVE    OF    A    DIFFERENTIAL   EQUATION.  5 

side.  Starting  from  any  position  in  the  plane,  the  point  P 
thus  moving  must  describe  a  branch  of  an  hyperbola  having 
the  two  axes  as  its  asymptotes;  accordingly,  the  complete 
integrar;rjj/  =  6'  is  the  equation  of  the  system  consisting  of 
these  hyperbolas. 

Prob.  5.  Write  the  differential  equation  which  requires  P  to  move 
in  a  direction  always  perpendicular  to  OP^  and  thence  derive  the 
equation  of  the  system  of  curves  described. 

Ans.   f^  =  -^;  a:'+/  =  C 
dx  y 

Prob.  6.  What  is  the  system  described  when  0  is  the  comple- 
ment of  ^?  Ans.  x^  —y"  =  C. 

Prob.  7.  If  0  =  26^,  show  geometrically  that  the  system  described 
•consists  of  circles,  and  find  the  differential  equation. 

Ans.  2xydx  =  {x^  —7")^. 

Art.  3.    Primitive  of  a  Differential  Equation. 

Let  us  now  suppose  an  ordinary  relation  between  x  and  y^ 
Avhich  may  be  represented  by  a  curve,  to  be  given.  By  differ- 
•entiation  we  may  obtain  an  equation  of  which  the  given  equa- 
tion is  of  course  a  solution  or  particular  integral.  But  by 
combining  this  with  the  given  equation  any  number  of  differ- 
ential equations  of  which  the  given  equation  is  a  solution  may 
be  found.     For  example,  from 

y  =  m{x  —  a)  (l) 

we  obtain  directly 

2ydy  =  nidx,  (2) 

of  which  equation  (i)  is  an  integral;  again,  dividing  (2)  by  (i) 
Ave  have 

2dy         dx 

and  of  this  equation  also  (i)  is  an  integral. 

If  in  equation  (i)  m  be  regarded  as  an  arbitrary  parameter, 
it  is  the  equation  of  a  system  of  parabolas  having  a  common 
axis  and  vertex.  The  differential  equation  (3),  which  does  not 
-contain  m,  is  satisfied  by  every  member  of  this  system  of  curves. 


0  DIFFERENTIAL    EQUATIONS. 

Hence  equation  (i)  thus  regarded  is  the  complete  integral  of 
equation  (3),  as  will  be  found  by  solving  the  equation  in  which 
the  variables  are  already  separated. 

Now  equation  (3)  is  obviously  the  only  differential  equatior* 
independent  of  m  which  could  be  derived  from  (i)  and  (2),  since 
it  is  the  result  of  eliminating  ;«.  It  is  therefore  the  "  differ- 
ential equation  of  the  system  ; "  and  in  this  point  of  view  the 
integral  equation  (i)  is  said  to  be  its  "primitive." 

Again,  if  in  equation  (i)  ^  be  regarded  as  the  arbitrary  con- 
stant, it  is  the  equation  of  a  system  of  equal  parabolas  having 
a  common  axis.  Now  equation  (2)  which  does  not  contain  a 
is  satisfied  by  every  member  of  this  system  of  curves;  hence  it 
is  the  differential  equation  of  the  system,  and  its  primitive  is 
equation  (i)  with  a  regarded  as  the  arbitrary  constant. 

Thus,  a  primitive  is  an  equation  containing  as  well  as  x  and 
y  an  arbitrary  constant,  which  we  may  denote  by  Cy  and  the 
corresponding  differential  equation  is  a  relation  between  x,  y^ 
and/,  which  is  found  by  differentiation,  and  elimination  of  C  if 
I  necessary.  This  is  therefore  also  a  method  of  verifying  the  com- 
plete integral  of  a  given  differential  equation.  For  example,  in 
verifying  the  complete  integral  (7)  in  Art.  I  we  obtain  by  differ- 
entiation I  =  —  Cp,  If  we  use  this  to  eliminate  C  from  equa- 
tion (7)  the  result  is  equation  (6);  whereas  the  process  before 
employed  was  equivalent  to  eliminating  /  from  equation  (6)> 
thereby  reproducing  equation  (7). 

Prob.  8.  Write  the  equation  of  the  system  of  circles  in  Prob.  7^ 
Art.  2,  and  derive  the  differential  equatioa  from  it  as  a  primitive. 

Prob.  9.  Write  the  equation  of  the  system  of  circles  passing 
through  the  points  (o,  b)  and  (o,  —  b)^  and  derive  from  it  the  differ- 
ential equation  of  the  system. 

Art.  4.    Exact  Differential  Equations. 

In  Art.  I  the  case  is  mentioned  in  which  Mdx  -\-  Ndy  is  an 
*'  exact  differential,"  that  is,  the  differential  of  a  function  of  x 
andjj/.     Let  M  denote  this  function;  then 

du  =  Mdx  +  Ndy,  (i) 


EXACT    DIFFERENTIAL    EQUATIONS,  7 

and  in  the  notation  of  partial  derivatives 

Then,  since  by  a  theorem  of  partial  derivatives  .  ^     =  _    .    , 

dM_dN 

dy   ~  dx-  ^^> 

This  condition  must  therefore  be  fulfilled  by  M  and  N  in 
order  that  equation  (i)  may  be  possible.  When  it  is  fulfilled 
Mdx  +  Ndy  =  o  is  said  to  be  an  "  exact  differential  equation," 
and  its  complete  integral  is 

u  =  C.  (3) 

For  example,  given  the  equation 

x{x  +  2y)dx  +  {x^  —  y^)dy  =  o, 

j^  =  x{x  +  27),  N  =  x"  —  y\  ——  =  2;r,  and  — —  =  2x ;  the 

condition  (2)  is  fulfilled,  and  the  equation  is  exact.  To  find  the 
function  u,  we  may  integrate  Mdx,  treating^/  as  a  constant;  thus, 

\x'+x'y=Y, 

in  which  the  constant  of  integration  Fmay  be  a  function  of  ^.*' 
The  result  of'differen'tiating  this  is 

x'^dx  -\-  2xy  dx  -\-  x^dy  =.  dY, 

which  should  be  identical  with  the  given  equation  ;  therefore, 
^F  =  y  <^,  whence  y=^y-{-Cy  and  substituting,  the  com- 
plete integral  may  be  written 

The  result  is  more  readily  obtained  if  we  notice  that  all 
terms  containing  x  and  dx  only,  or  y  and  dy  only,  are  exact 
differentials ;  hence  it  is  only  necessary  to  examine  the  terms 
containing  both  x  and  y.  In  the  present  case,  these  are 
2xy  dx  +  x'^dy,  which  obviously  form  the  differential  of  xy  ; 
whence,  integrating  and  multiplying  by  3,  we  obtain  the  result 
above. 

The  complete  integral  of  any  equation,  in  whatever  way  it 


8^  DIFFERENTIAL   EQUATIONS. 

was  found,  can  be  put  in  the  form  u  =  C,  by  solving  for  C» 
Hence  an  exact  differential  equation  du  =o  can  be  obtained, 
which  must  be  equivalent  to  the  given  equation 

Mdx  +  Ndy  =  o,  (4) 

here  supposed  not  to  be  exact.  The  exact  equation  du  =  o 
must  therefore  be  of  the  form 

fx{Mdx  +  Ndy)  =  o,  (5) 

where  //  is  a  factor  containing  at  least  one  of  the  variables  x 
and^.     Such  a  factor  is  called  an  **  integrating  factor"  of  the 
given  equation.       For  example,  the  result  of  differentiating 
equation  (7),  Art.  I,  when  put  in  the  form  u  =  C,  is 
(I  ^j;)dx  +  {i  +x)dy  _^ 

so  that  (i  —  jSy*  is  an  integrating  factor  of  equation  (6).  It 
is  to  be  noticed  that  the  factor  by  which  we  separated  the 
variables,  namely,  (i  —/)"'( i -f  ^)~S  is  also  an  integrating 
factor.  J   • 

It  follows  that  if  an  integrating  factor  can  be  discovered, 
the  given  differential  equation  can  at  once  be  solved.*  Such 
a  factor  is  sometimes  suggested  by  the  form  of  the  equation. 

Thus,  given  {y  —  x)dy-\-ydx  =  o, 

the  terms  ydx  —  xdy,  which  contain  both  x  and  y,  are  not  ex- 
act, but  become  so  when  divided  by  either  x''  or  y;  and  be- 
cause the  remaining  term  contains^  only,  j"'*  is  an  integrating 
factor  of  the  whole  expression.     The  resulting  integral  is 

^ogy  +  -=C 

Prob.  10.  Show  from  the  integral  equation  in  Prob.  9,  Art.  3,  that 
x~^  is  an  integrating  factor  of  the  differential  equation. 

Prob.  II.  Solve  the  equation  x(x^  +  $y^)dx  -\-y(y'  -f  3^^)^y  =  o- 

Ans.  X*  +  6::cV  +y  =  ^• 

*  Since  juM  and  //iVin  the  exact  equation  (5)  must  satisfy  the  condition  (2), 
we  have  a  partial  differential  equation  for  JU',  but  as  a  general  method  of  finding 
/£  this  simply  comes  back  to  the  solution  of  the  original  equation. 


HOMOGENEOUS   EQUATION.  V 

X  dy — y  dx 
Prob.  1 2.  Solve  the  equation  y  dy-\-xdx-^^ ^—^ — 5—.  =  o. 

^  ^  X   -\-  y 

x^  -^  v^  y 

(Ans.   ^-=^  +  tan-  -  =  ^.) 

r  2  X  ' 

Prob.  13.  If  u  =  c  is  a.  form  of  the  complete  integral  and  /^  the 
corresponding  integrating  factor,  show  that  )^f{u)  is  the  general 
expression  for  the  integrating  factors. 

Prob.  14.  Show  that  the  expression  x'^y^{fnydx  -\-  nxdy)  has  the 
integrating  factor  ;(-*'»-^-«y*«-'-P;  and  by  means  of  such  a  factor 
solve  the  equation  ^(y  +  2x*)dx  +  x{x*  —  2y^)dy  =■  o. 

Ans.   2x*y  —  y  =  cx^. 

Prob.  15.  Solve  (x"^  -{- y')dx  —  2xydy  —  o.       Ans.  x"^  —  y"^  =  ex. 


Art.  5.    Homogeneous  Equation. 

The  differential  equation  Mdx  +  Ndy  =  o   is  said  to   be 

homogeneous  when  M  and  N  are  homogeneous  functions  of 

X  and  y  of  the  same  degree ;  or,  what  is  the  same  thing,  when 

dy  y 

-r-  is  expressible  as  a  function  of  — .     If  in  such  an  equation 

the  variables  are  changed  from  x  and  j  to  ;ir  and  v,  where 

y 

v  =z  —]         whence    y  =z  xv     and     dy  =  xdv  -\-  vdx, 

the  variables  x  and   v  will  be  separable.     For  elxample,  the 

equation 

{x  —  2y)dx  +  ydy  =  o 

is    homogeneous ;    making   the   substitutions    indicated    and 
dividing  by  x, 

(i  —  2v)dx  -\-  v{xdv  +  vdx)  =  o, 

dx  vdv 

whence  1- 


X    '   (^  —  i)' 
log 
and  restoring  J, 

The  equation  Mdx  -\-  Ndy  =  o  can  always  be  solved  when 


Integrating,  log  x  +  log  {v  —  i)  —  ^^— -y  =  C; 


log  (y  —  ^) =  ^• 

*=>  ^-^         ^      y  —  X 


10  DIFFERENTIAL   EQUATIONS. 

M  and  N  are  functions  of  the  first  degree,  that  is,  when  it  is 
of  the  form 

^  (ax  +  by  +  c)dx  +  {a'x  +  b'y  +  c')dy  =  o. 

FfJf,  assuming  x  =  x'  -\-  //,  y  =  /  -\-  k,  it  becomes 
{ax'+  dy+  ak  +  dk+c)dx'+{a'x'+  b'y'-\-a'h^b'k-\-c')dy' =o, 
which,  by  properly  determining  //  and  k,  becomes 

{ax'  +  by')dx'  +  (a'x'  +  b'y')dy\ 
a  homogeneous  equation. 

This  method  fails  when  a\  b  =  a' :  b',  that  is,  when  the 
equation  takes  the  form 

{ax  -{■  by  -\-  c)dx  -\-  [m{ax  +  by)  +  c'']dy  =  o ; 

but  in  this  case  if  we  put  z  =.  ax  -\-  by,  and  eliminate  y,  it  will 
be  found  that  the  variables  x  and  z  can  be  separated. 

Prob.  1 6.  Show  that  a  homogeneous  differential  equation  repre- 
sents a  system  of  similar  and  similarly  situated  curves,  the  origin 
being  the  center  of  similitude,  and  hence  that  the  coniplete  integral 
may  be  written  in  a  form  homogeneous  in  x.y,  and  c. 

Prob.  17.  Solve  xdy  —  y  dx  —  ^{x^  -\-  y^)dx  —  o. 

Ans.  ^'  =  ^'  —  2cy, 

Prob.  18.  Solve  (3^  —  7^  +  i)dx  +  (7;^  —  3-^  +  3)^  =  o. 

Ans.  {y  —  x-\-  \)\y-\-  x  —  if  —  c. 

Prob.  19.  Solve  {x^  -\-  y^)dx  —  2xydy  =  o.      Ans.  x"  — y  =  ex. 

Prob.  20.  Solve  (i  +  xy)ydx  +  (i  —  xy)xdy  =  o  by  introducing 

the  new  variable  z  =  xy.  Ans.  x  =  Cye'^y. 

Prob.  21.  Solve  ^=«^+^j'+^.       Ans.  abx-^b''y-^a-^bc  =  Ce^*, 

Art.  6.    The  Linear  Equation. 

A  differential  equation  is  said  to  be  **  linear  "  when  (one  of 
the  variables,  say  x,  being  regarded  as  independent,)  it  is  of 
the  first  degree  with  respect  to  j,  and  its  derivatives.  The 
linear  equation  of  the  first  order  may  therefore  be  written  in 
the  form 


THE    LINEAR    EQUATION.  11 

where  P  and  Q  are  functions  of  x  only.  Since  the  second 
member  is  a  function  of  x,  an  integrating  factor  of  the  first 
member  will  be  an  integrating  factor  of  the  equation  provided 
it  contains  x  only.     To  find  such  a  factor,  we  solve  the  equation 

|  +  />  =  o,  (.) 

dy 
which  is  done  by  separating  the  variables;  thus,  —  =  ~  Pdx ; 


whence  \o%y  ^  c  —  I  Pdx  or 


y  =  ^-"^"'.  (3) 

Putting  this  equation  in  the  form  u  =^  Cy  the  corresponding 
exact  equation  is 

e^^^\dy  +  Pydx)  =  o, 

whence   e'    '^   is  the  integrating  facto'  required.     Using  this 
factor,  the  general  solution  of  equation  (i)  is 

eS^'^y  ^J/p^^Qdx  +  C,  (4> 

In  a  given  example  the  integrating  factor  should  of  course 
be  simplified  in  form  if  possible.     Thus 

(i  +  x^^dy  ^  {m-\^  xy)dx 

is  a  linear  equation  for  j/ ;  reduced  to  the  form  (i),  it  is 

dy  X  m 


7^^  = 


dx        i+x'-"        I  +  x'' 
from  which 

X  dx  I 

+  X'  ^   ~  2 

The  integrating  factor  is,  therefore. 


/^  T  p  X  dx  I  ,       , 


fPdx  I 

whence  the  exact  equation  is 

dy  xy  dx  mdx 


4/(1+^')       (i+;.')!-(i+4rr 


12  DIFFERENTIAL    EQUATIONS. 

Integrating,  there  is  found 

y       _       ^-^       .  r 

or 

y  =  mx  -\-  C  \^{l  -\-  x^). 

An  equation  is  sometimes  obviously  linear,  not  for  j,  but 

for  some  function  of  y.     For  example,  the  equation 

dy  , 

-r-  -T  tan  y  ^=  X  sec  y 

when  multiplied  by  cos  y  takes  a  form   linear  for  sin  y ;  the 
integrating  factor  is  ^,  and  the  complete  integral 

s\n  y  =^  X  —  I  -\-  ce  ~*. 

dy 
In  particular,  the  equation  -j-  -^  Py  =  Qy""^  which  is  known  as 

"  the  extension  of  the  linear  equation,"  is  readily  put  in  a  form 
linear  for  y". 

dy  1 

Prob.  22.  Solve  x"—  +  (i  —  2x)y  =  x*.       Ans.  y  =  ^'(i  +  ce''). 

Prob.  23.  Solve  cos  x-^  -\-y  —  i  +  sin  ^  =  o. 

Ans.  jr(sec  x  +  tan  x)  =  x  -\-  c. 

Ans.  y  =  sin  X  -^  c  cos  x. 
Prob.  25.  Solve  -r-  =  ^V  —  xy.         Ans.   -  =:'jc'  -}-  i  -f-^^'. 

Prob.  26.  Solve  ^'  = ^^^-^.        Ans.  -  =  ^  -/+  ^e-y\ 

dx      xy  -\-  xy  x 

Art.  7.    First  Order  and  Second  Degree. 

If  the  given  differential  equation  of  the  first  order,  or  re- 
lation between  x,  y,  and  p,  is  a  quadratic  for  p,  the  first  step 
in  the  solution  is  usually  to  solve  for  p.  The  resulting  value 
of  /  will  generally  involve  an  irrational  function  of  ;r  and  y; 
so  that  an  equation  expressing  such  a  value  of  /,  like  some  of 
those  solved  in  the  preceding  pages,  is  not  properly  to  be  re- 


dy 
Prob.  24.  Solve  —  cos  x  -{-  y  s'm  x  =  i. 

uX 


FIRST    ORDER    AND    SECOND    DEGREE.  13 

garded  as  an  equation  of  the  first  degree.  In  the  exceptional 
case  when  the  expression  whose  root  is  to  be  extracted  is  a 
perfect  square,  the  equation  is  decomposable  into  two  equa- 
tions properly  of  the  first  degree.  For  example,  the  equation 
4rXl+4/)=2/(x'+/) 

y  X     » 

when  solved  for  p  gives  2p  —  -,  or  2p  — —\'  it   may  therefore 

be  written  in  the  form 

{2px  -  y){2py  -x)  =  o, 
and  is  satisfied  by  putting  either 

dy  _  y  dy  _  X 

dx       2x'  dx      2y 

The  integrals  of  these  equations  are 

y""  =  ex     and     2y''  —  x^  =  d 

and  these  form  two  entirely  distinct  solutions  of  the  given 
equation. 

As  an  illustration  of  the  general  case,  let  us  take  the  equation 

./=^,      or      %=±^.  (I) 

Separating  the  variables  and  integrating, 

Vx  ±Vy=  ±Vc,  (2) 

and  this  equation  rationalized  becomes 

(x-yy-2c{x+y)+c'  =  0.  (3) 

There  is  thus  a  single  complete  integral  containing  one  arbi- 
trary constant  and  representing  a  single  system  of  curves; 
namely,  in  this  case,  a  system  of  parabolas  touching  each  axis 
at  the  same  distance  c  from  the  origin.  The  separate  Equa- 
tions given  in  the  form  (2)  are  merely  branches  of  the  same 
parabola. 

Recurring  now  to  the  geometrical  interpretation  of  a  differ- 
ential equation,  as  given  in  Art.  2,  it  was  stated  that  an  equa- 
tion of  the  first  degree  determines,  in  general,  for  assumed 
values  of  x  and  y,  that  is,  at  a  selected  point  in  the  plane,  a 
single  value  of  p.     The  equation  was,  of  course,  then  supposed 


14  DIFFERENTIAL   EQUATIONS. 

rational  in  x  and  y.*  The  only  exceptions  occur  at  points  for 
which  the  value  of  p  takes  the  indeterminate  form  ;  that  is, 
the  equation  being  Mdx  -f-  Ndy  =  o,  at  points  (if  any  exist) 
for  which  M  =  o  and  iV  =  o.  It  follows  that,  except  at  such 
points,  no  two  curves  of  the  system  representing  the  complete 
integral  intersect,  while  through  such  points  an  unlimited  num- 
ber of  the  curves  may  pass,  forming  a  "pencil  of  curves."  f 

On  the  other  hand,  in  the  case  of  an  equation  of  the  second 
degree,  there  will  in  general  be  two  values  of  p  for  any  given 
point.  Thus  from  equation  (i)  above  we  find  for  the  point 
{4,  i),  /  =  ±  J;  there  are  therefore  two  directions  in  which  a 
point  starting  from  the  position  (4,  i)  may  move  while  satis- 
fying the  differential  equation.  The  curves  thus  described 
represent  two  of  the  particular  integrals.  If  the  same  values 
of  X  and  y  be  substituted  in  the  complete  integral  (3),  the  re- 
sult is  a  quadratic  for  c,  giving  c  =■  g  and  ^  =  i,  and  these 
determine  the  two  particular  integral  curves,  Vx-\-  Vy  =  $, 
and   Vx  —  Vy  =  1. 

In  like  manner  the  general  equation  of  the  second  degree, 
which  may  be  written  in  the  form 

Lf  +  Mp-{-N=o, 
where  L,  M,  and  TV  are  one-valued  functions  .of  x  and  y,  repre- 
sents a  system  of  curves  of  which  two  intersect  in  any  given 
point  for  which  p  is  found  to  have  two  real  values.  For  these 
points,  therefore,  the  complete  integral  should  generally  give 
two  real  values  of  c.  Accordingly  we  may  assume,  as  the 
standard  form  of  its  equation, 

Pc'  +  Qc  +  R  =  o, 

*  In  fact  /  was  supposed  to  be  a  one-valued  function  of  x  and  y;  thus, 
p  =  sin-*;c  would  not  in  this  connection  be  regarded  as  an  equation  of  the  first 
degree. 

f  In  Prob.  6,  Art.  3,  the  integral  equation  represents  the  pencil  q^  circles  pass- 
ing through  the  points  (o,  ^)  and  (o,  —  ^);  accordingly/  in  the  differential  equa- 
tion is  indeterminate  at  these  points.  In  some  cases,  however,  such  a  point  is 
merely  a  node  of  one  particular  integral.  Thus  in  the  illustration  given  in  Art.  2, 
P  is  indeterminate  at  the  origin,  and  this  point  is  a  node  of  the  only  particular 
integral,  xy  =  o,  which  passes  through  it. 


SINGULAR   SOLUTIONS.  15 

where  P,  Q,  and  R  are  also  one-valued  functions  of  x  and  j/. 
If' there  are  points  which  make  p  imaginary  in  the  differential 
equation,  they  will  also  make  c  imaginary  in  the  integral. 

Prob.  27.  Solve  the  equation  /'  +y  =  i  and  reduce  the  inte- 
gral to  the  standard  form. 

Ans.  (y  +  cos  x)c''  —  2^  sin  ^  +  jf  —  cos  ^  =  o. 

Prob.  28.  Solve  j^/"*  +  2xJ>  —  y  =  o,  and  sHow  that  the  intersect- 
ing curves  at  any  given  point  cut  at  right  angles. 

Prob.  29.  Solve  (x^  -{-  i)j>''  =  i.  Ans.  ^V  —  zcxe^  =  i. 

Art.  8.    Singular  Solutions. 

A  differential  equation  not  of  the  first  degree  sometimes 
admits  of  what  is  called  a  "  singular  solution  ;  "  that  is  to  say,  a 
solution  which  is  not  included  in  the  complete  integral.  For 
suppose  that  the  system  of  curves  representing  the  complete 
integral  has  an  envelope.  Every  point  A  of  this  envelope 
is  a  point  of  contact  with  a  particular  curve  of  the  complete  in- 
tegral system  ;  therefore  a  point  moving  in  the  envelope  when 
passing  through  A  has  the  same  values  of  ;ir,  j,  and /as  if  it 
were  moving  through  A  in  the  particular  integral  curve.  Hence 
such  a  point  satisfies  the  differential  equation  and  will  continue 
to  satisfy  it  as  long  as  it  moves  in  the  envelope.  The  equation 
of  the  envelope  is  therefore  a  solution  of  the  equation. 

As  an  illustration,  let  us  take  the  system  of  straight  lines 
whose  equation  is 

:y  =  ^^  +  p  (I) 

where  c  is  the  arbitrary  parameter.  The  differential  equation 
derived  from  this  primitive  is 

^=/;r+-^,  (2) 

of  which  therefore  (i)  is  the  complete  integral. 

Now  the  lines  represented  by  equation  (i),  for  different 
values  of  c^  are  the  tangents  to  the  parabola 

/  =  \ax,  (3) 


16 


DIFFERENTIAL    EQUATIONS. 


A  point  moving  in  this  parabola  has  the  same  value  of/  as  if  it 

were  moving  in  one  of  the  tan- 
gents, and  accordingly  equation 
(3)  will  be  found  to  satisfy  the 
differential  equation  (2). 

It  will  be  noticed  that  for 
any  point  on  the  convex  side  of 
the  parabola  there  are  two  real 
values  of  p\  for  a  point  on  the 
other  side  the  values  of  p  are 
imaginary,  and  for  a  point  on 
the  curve  they  are  equal.  Thus 
its  equation  (3)  expresses  the 
relation  between  x  and  y  which  must  exist  in  order  that  (2) 
regarded  as  a  q^uadratic  for  p  may  have  equal  roots,  as  will  be 
seen  on  solving  that  equation. 

In  general,  writirig  the  differential  equation  in  the  form 

Z/  +  J//  +  i\^  =  o,  (4) 

the  condition  of  equal  roots  is 

M'-^LN  =  o.  "  (5> 

The  first  member  of  this  equation,  which  is  the  "  discrimi- 
nant "  of  equation  (4),  frequently  admits  of  separation  into 
factors  rational  in  x  and  j.  Hence,  if  there  be  a  singular  solu- 
tion, its  equation  will  be  found  by  putting  the  discriminant  o£ 
the  differential  equation,  or  one  of  its.  factors,  equal  to  zero. 

It  does  not  follow  that  every  such  equation  represents  a  solu- 
tion of  the  differential  equation.  It  can  only  be  inferred  that 
it  is  a  locus  of  points  for  which  the  two  values  of  p  become 
equal.  Now  suppose  that  two  distinct  particular  integral 
curves  touch  each  other.  At  the  point  of  contact,  the  two 
values  of/,  usually  distinct,  become  equal.  The  locus  of  such 
points  is  called  a  "tac-locus."  Its  equation  plainly  satisfies  the 
discriminant,  but  does  not  satisfy  the  differential  equation.  An 
illustration  is  afforded  by  the  equation 


SINGULAR    SOLUTIONS.  17 

of  which  the  complete  integral  is  y  +  (-^  —  ^Y  =  a*,  and  the 
discriminant,  see  equation  (5),  is_^'(y  —  a"^)  =  o.  \ 

This  is  satisfied  by  jy  =  a,  jy  =  —a,  and  jj/  =  o,  the  first  two 
of  which  satisfy  the  differential  equation,  while  /  =  o  does  not.     '^ 
The  complete  integral  represents  in  this  case  all  circles  of  radius  ' 

a  with  center  on  the  axis  of  x.  Two  of  these  circles  touch  at 
every  point  of  the  axis  of  x^  which  is  thus  a  tac-locus,  while 
y  ^  a  and  y  =z  —  a  constitute  the  envelope. 

The  discriminant  is  the  quantity  which  appears  under  the 
radical  sign  when  the  general  equation  (4)  is  solved  for/,  and 
therefore  it  changes  sign  as  we  cross  the  envelope.  But  the 
values  of  /  remain  real  as  we  cross  the  tac-locus,  so  that  the 
discriminant  cannot  change  sign.  Accordingly  the  factor  which 
indicates,  a  tac-locus  appears  with  an  even  exponent  (as  y"^  in 
the  example  above),  whereas  the  factor  indicating  the  singular 
solution  appears  as  a  simple  factor,  or  with  an  odd-.exponent. 

A  simple  factor  of  the  discriminant,  or  one  with  an  odd  ex- 
ponent, gives  in  fact  always  the  boundary  between  a  region  of 
the  plane  in  which/  is  real  and  one  in  which/  is  imaginary; 
nevertheless  it  may  not  give  a  singular  solution.  For  the  two 
arcs  of  particular  integral  curves  which  intersect  in  a  point  on 
the  real  side  of  the  boundary  may,  as  the  point  is  brought  up 
to  the  boundary,  become  tangent  to  each  other,  but  not  to  the 
boundary  curve.  In  that  case,  since  they  cannot  cross  the 
boundary,  they  become  branches  of  the  same  particular  inte- 
gral forming  a  cusp.  A  boundary  curve  of  this  character  is 
called  a  "cusp-locus"  ;  the  value  of  /  for  a  point  moving  in  it 
is  of  course  different  from  the  equal  values  of  /  at  the  cusp,  and 
therefore  its  equation  does  not  satisfy  the  differential  equation.* 

Prob.  30.  To  what  curve  is  the  line  y  =  mx  -\-  a  |/(i  —  m^) 
always  tangent  ?  Ans.  y  —  j^:'  =  ^^ 

Prob.  31.  Show  that  the  discriminant  of  a  decomposable  differ- 

*  Since  there  is  no  reason  \y-hy  the  values  of/  referred  to  should  be  identical, 
we  conclude  that  the  equation  Lp^  +  Mp  -\-  N=o  has  not  in  general  a  singular 
solution,  its  discriminant  representing  a  cusp-locus  except  when  a  certain  con- 
dition is  fulfilled. 


18  DIFFERENTIAL   EQUATIONS. 

cntial  equation  cannot  be  negative.  Interpret  the  result  of  equating 
it  lo  zero  in  the  illustrative  example  at  the  beginning  of  Art.  7. 

Prob.  32.  Show  that  the  singular  solutions  of  a  homogeneous  dif- 
ferential equation  represent  straight  lines  passing  through  the  origin. 

Prob.  33.  Solve  the  equation  xp^  —  2yp  -\-ax  =  o. 

Ans.  x"  —  2cy  -\-  ac"  ■=  o  \  singular  solution  y"  —  ax^, 

Prob.  34.  Show  that  the  equation /'  +  2xp  —  y=zo  has  no  sin- 
gular solution,  but  has  a  cusp-locus,  and  that  the  tangent  at  every 
cusp  passes  through  the  origin. 

Art.  9.    Singular  Solution  from  Complete  Integral. 

When  the  complete  integral  of  a  differential  equation  of 
the  second  degree  has  been  found  in  the  standard  form 

V  •     Pc^^Qc  +  R  =  o  (I) 

(see  the  end  of  Art.  7),  the  substitution  of  special  values  of  x 
and  y  in  the  functions  P,  Q,  and  R  gives  a  quadratic  for  c  whose 
roots  determine  the  two  particular  curves  of  the  system  which 
pass  through  a  given  point.  If  there  is  a  singular  solution, 
that  is,  if  the  system  of  curves  has  an  envelope,  the  two 
curves  which  usually  intersect  become  identical  when  the  given 
point  is  moved  up  to  the  envelope.  Every  point  on  the  en- 
velope therefore  satisfies  the  condition  of  equal  roots  for  equa- 
tion (i),  which  is 

Q^-4PJ^  =  o;  (2) 

and,  reasoning  exactly  as  in  Art.  8,  we  infer  that  the  equation 
of  the  singular  solution  will  be  found  by  equating  to  zero  the 
discriminant  of  the  equation  in  c  or  one  of  its  factors.  Thus 
the  discriminant  of  equation  (i),  Art.  8,  or  "^-discriminant,"  is 
the  same  as  the  "/-discriminant,"  namely,  y  —  4ax,  which 
equated  to  zero  is  the  equation  of  the  envelope  of  the  system  of 
straight  lines. 

But,  as  in  the  case  of  the  /-discriminant,  it  must  not  be 
inferred  that  every  factor  gives  a  singular  solution.  For  ex- 
ample, suppose  a  squared  factor  appears  in  the  ^-discriminant. 
The  locus  on  which  this  factor  vanishes  is  not  a  curve  in  cross- 
ing which  c  and/  become  imaginary.     At  any  point  of  it  there 


SINGULAR  SOLUTION  FROM  COMPLETE  INTEGRAL.  19 

Avill  be  two  distinct  values  of  p,  corresponding  to  arcs  of  par- 
ticular integral  curves  passing  through  that  point ;  but,  since 
there  is  but  one  value  of  c,  these  arcs  belong  to  the  same  par- 
ticular integral,  hence  the  point  is  a  double  point  or  node. 
The  locus  is  therefore  called  a  ''node-locus."  The  factor  repre- 
senting it  does  not  appear  in  the  /-discriminant,  just  as  that 
representing  a  tac-locus  does  not  appear  in  the  ^-discriminant. 

Again,  at  any  point  of  a  cusp-locus,  as  shown  at  the  end  of 
Art.  8,  the  two  branches  of  particular  integrals  become  arcs  ot 
the  same  particular  integral ;  the  values  of  c  become  equal,  so 
that  a  cusp-locus  also  makes  the  <;-discriminant  vanish. 

The  conclusions  established  above  obviously  apply  also  to 
equations  of  a  degree  higher  than  the  second.  In  the  case  of 
the  ^-equation  the  general  method  of  obtaining  the  condition 
for  equal  roots,  which  is  to  eliminate  c  between  the  original  and 
the  derived  equation,  is  the  same  as  the  process  of  finding  the 
envelope  or  *'  locus  of  the  ultimate  intersections  "  of  a  system 
of  curves  in  which  c  is  the  arbitrary  par^ameter. 

Now  suppose  the  system  of  curves  to  have  for  all  values  of 
c"^  2,  double  point,  it  is  obvious  that  among  the  intersections 
of  two  neighboring  curves  there  are  two  in  the  neighborhood 
of  the  nodes,  and  that  ultimately  they  coincide  with  the  node, 
which  accounts  for  the  node-locus  appearing  twice  in  the  dis- 
criminant or  locus  of  ultimate  intersections.     In  like  manner, 

*  It  is  noticed  in  the  second  foot-note  to  Art.  7  that  for  an  equation  of  the 
first  degree  p  takes  the  indeterminate  form,  not  only  at  a  point  through  which  all 
curves  of  the  system  pass  (where  the  value  of  c  would  also  be  found  indeter- 
minate), but  at  a  node  of  a  particular  integral.  So  also  when  the  equation  is  of 
the  «th  degree,  if  there  is  a  node  for  a  particular  value  of  c,  the  n  values  of  c  at 
the  point  (which  is  not  on  a  node-locus  where  two  values  of  c  are  equal)  deter- 
mine n  +  I  arcs  of  particular  integrals  passing  through  the  point ;  and  there- 
fore there  are  w  +  i  distinct  values  of  /  at  the  point,  which  can  only  happen 
when  p  takes  the  indeterminate  form,  that  is  to  say,  when  all  the  coefficients  of 
the /-equation  (which  is  of  the  wth  decree)  vanish.  See  Cayley  on  Singjular  So- 
lutions in  the  Messenger  of  Mathematics.  New  Series,  Vol.  II,  p.  10  (Collected 
Mathematical  Works,  Vol.  VIlI,  p.  529).  The  present  tneory  of  Singular  Solu- 
tions was  established  by  Cayley  in  this  paper  and  its  continuation.  Vol.  VI,  p.  33. 
See  also  a  paper  by  Dr.  Glaisher,  Vol.  XII,  p.  i. 


20  DIFFERENTIAL   EQUATIONS. 

if  there  is  a  cusp  for  all  values  of  ^,  there  are  three  intersections 
of  neighboring  curves  (all  of  which  may  be  real)  which  ulti- 
mately coincide  with  the  cusp ;  therefore  a  cusp-locus  will 
appear  as  a  cubed  factor  in  the  discriminant.* 

'  Prob.  35.  Show  that  the  singular  solutions  of  a  homogeneous 
equation  must  be  straight  lines  passing  through  the  origin. 

Prob.  36.  Solve  3^'/  —  "^xyp  +  4>''  —  a:'  =  o,  and  show  that  there 
is  a  singular  solution  and  a  tac-focus. 

Prob.  37.  Solve  yf-\-  2xp  —  ^  =  o,  and  show  that  there  is  an 
imaginary  singular  solution.  Ans.  f'  =  2cx  -\-  c^. 

Prob.  38.  Show  that  the  equation  (i  —  x^)p'  =  i  —  y  represents 
a  system  of  conies  touching  the  four  sides  of  a  square. 

Prob.  39.  Solve  yp^  —  4xp-\-y  =  o  ;  examine  and  interpret  both 
discriminants.  Ans.  ^'  +  2cx(^'^  —  8^')  —  3^y  -\-y'  =  o. 


Art.  10.    Solution  by  Differentiation. 

The  result  of  differentiating  a  given  differential  equation  of 
the  first  order  is  an  equation  of  the  second  order,  that  is,  it 

contains  the  derivative  -y-^ ;  but,  if  it  does  not  contain  y>  ex- 
plicitly, it  may  be  regarded  as  an  equation  of  the  first  order  for 
the  variables  x  and  p.  If  the  integral  of  such  an  equation  can 
be  obtained  it  will  be  a  relation  between  x,  /,  and  a  constant 
of  integration  c,  by  means  of  which  p  can  be  eliminated  from 
the  original  equation,  thus  giving  the  relation  between  x,  j, 
and  c  which  constitutes  the  complete  integral.  For  example, 
the  equation 

J  +  2^J/=.^+/,  (I) 

*  The  discriminant  of  Pc*  -\-  Qc  ^  /^  =  o  represents  in  general  an  envelope, 
no  further  condition  requiring  to  be  fulfilled  as  in  the  case  of  the  discriminant 
of  Lp^  +  Mp  -\-  JV  =  o.  Compare  the  foot-note  to  Art.  8.  Therefore  where 
there  is  an  integral  of  this  form  there  is  generally  a  singular  solution,  although 
Lp^  +  Mp  -|-  A''  =  o  has  not  in  general  a  singular  solution.  We  conclude,  there- 
fore, that  this  equation  (in  which  Z,  M,  and  iV  are  one- valued  functions  of  x 
and  y)  has  not  in  general  an  integral  of  the  above  form  in  which  P,  Q,  and  /? 
are  one-valued  functions  of  x  and  y.  Cayley,  Messenger  of  Mathematics,  New 
Series,  Vol.  VI,  p.  23. 


SOLUTION    BY    DIFFERENTIATION.  21 

when  solved  for^,  becomes 

y  =  X+Vj)  (2) 

whence  by  differentiation 

£^  dp 

2  V/ 

The  variables  can  be  separated  in  this  equation,  and  its  inte- 
gral is 


/='+r^^I-  (3) 


^^  =  z- 


Substituting  in  equation  (2),  we  find 

which  is  the  complete  integral  of  equation  (i). 

This  method  sometimes  succeeds  with  equations  of  a  higher 
■degree  when  the  solution  with  respect  to  p  is  impossible  or 
leads  to  a  form  which  cannot  be  integrated.  A  differential 
equation  between  p  and  one  of  the  two  variables  will  be  ob- 
tained by  direct  integration  when  only  one  of  the  variables  is 
explicitly  present  in  the  equation,  and  also  when  the  equation 
is  of  the  first  degree  with  respect  to  x  and  y.  In  the  latter 
case  after  dividing  by  the  coefficient  of  j^,  the  result  of  differ- 
•entiation  will  be  a  linear  equation  for  ;ir  as  a  function  of  ^,  so 
that  an  expression  for  x  in  terms  of  /)  can  be  found,  and  then 
by  substitution  in  the  given  equation  an  expression  for  y  in 
terms  of  p.  Hence,  in  this  case,  any  number  of  simultaneous 
values  of  x  and  y  can  be  found,  although  the  elimination  of  p 
may  be  impracticable. 

In  particular,  a  homogeneous  equation  which  cannot  be 
•solved  for  ;>  may  be  soluble  for  the  ratio  /  :  ;tr,  so  as  to  assume 
the  form  /  =  x(f)(p).     The  result  of  differentiation  is 

in  which  the  variables  x  and  p  can  be  separated. 
Another  special  case  is  of  the  form 

y  =  Ar  +/(»,  (I) 


22  DIFFERENTIAL   EQUATIONS. 

which  is  known  as  Clairaut's  equation.     The  result  of  differ- 
entiation is 

which  implies  either 

•^+/V)=o,         or        ^  =  o. 

The  elimination  of  p  from  equation  (i)  by  means  of  the 
first  of  these  equations  *  gives  a  solution  containing  no  arbi- 
trary constant,  that  is,  a  singular  solution.  The  second  is  a 
differential  equation  for  p ;  its  integral  is  /  =  ^,  which  in 
equation  (i)  gives  the  complete  integral 

y  =  cx^Ac)-  (2> 

This  complete  integral  represents  a  system  of  straight  lines, 
the  singular  solution  representing  the  curve  to  which  they  are 
all  tangent.     An  example  has  already  been  given  in  Art.  8. 

A  differential  equation  is  sometimes  reducible  to  Clairaut's 

form  by  means  of  a  more  or  less  obvious  transformation  of  the 

variables.     It  may  be  noticed  in  particular  that  an  equation  of 

the  form 

y=nxp-\-  (f){x,  p) 

is  sometimes  so  reducible  by  transformation  to  the  independent 
variable  z^  where  x  =  z*"  \  and  an  equation  of  the  form 

y  =  nxp+  (p{y,p), 
by  transformation  to  the   new  dependent  variable  v  =  y"".     A 
double   transformation    of    the   form  indicated    may   succeed 
when  the  last  term  is  a  function  of  both  x  and  j' as  well  as  of/. 

Prob.  40.  Solve  the  equation  ^y  =  2/'  +  3/';  find  a  singular 
solution  and  a  cusp-locus.  Ans.  (x-\-y  -\-  c  —  -J)"  =  -{x-\-cy, 

Prob.  41.  Solve  2y  =  xp  -\ — ,  and  find  a  cusp-locus. 

P 
Ans.  «V'  —  i2acxy  +  8^/  —  i2^y  +  i6ax^  =  o. 

*  The  equation  is  in  fact  the  same  that  arises  in  the  general  method  for  the 
condition  of  equal  roots.     See  Art.  9. 


GEOMETRIC    APPLICATIONS  ;     TRAJECTORIES.  2^ 

Prob.  42.   Solve  (x^  —  a^)p^  ^  2xyp  -\-  y^  —  a^  =  o. 

Ans.  The  circle  x^  -\- y"  =  a^^  and  its  tangents. 

Prob.  43.  Solve  y—  —  ^/  +  x'p^, 

Ans.  c^x  -\-  c  —  xy  =  o,  and  i  +  4^^  =  o-    > 

Prob.  44.   Solve  /  —  4xyp  +  8/  =  o. 

Ans.  y  =  c{x  —  cY  \  2'jy  =  4^'  and_y  =  o  are  singulai 
solutions; 7  =  o  is  also  a  particular  integral. 

Prob.  45.  Solve  x'^(y  —  px)  —yp"".  Ans.  y  =  cx"^  +  c". 

Art.  11.    Geometric  Applications  ;  Trajectories. 

Every  property  of  a  curve  which  involves  the  direction  of 
its  tangents  admits  of  statement  in  the  form  of  a  differential 
equation.  The  solution  of  such  an  equation  therefore  deter- 
mines the  curve  having  the  given  property.  Thus,  let  it  be 
required  to  determine  the  curve  in  which  the  angle  between 
the  radius  vector  and  the  tangent  is  n  times*  the  vectorial 
angle.  Using  the  expression  for  thb  trigonometric  tangent  of 
that  angle,  the  expression  of  the  property  in  polar  coordi- 
nates is  tx^  ^   -  -^ 

——  =  tan  nd, 
dr 

Separating  the  variables  and  integrating,  the  complete 
integral  is 

r"  =  c^  sin  nd. 

The  mode  in  which*  the  constant  of  integration  enters  here 
shows  that  the  property  in  question  is  shared  by  all  the  mem- 
bers of  a  system  of  similar  curves. 

The  solution  of  a  question  of  this  nature  will  thus  in  gen- 
eral be  a  system  of  curves,  the  complete  integral  of  a  differential 
equation,  but  it  may  be  a  singular  solution.  Thus,  if  we  ex- 
press the  property  that  the  sum  of  the  intercepts  on  the  axes 
made  by  the  tangent  to  a  curve  is  equal  to  the  constant  a,  the 
straight  lines  making  such  intercepts  will  themselves  consti- 
tute the  complete  integral  system,  and  the  curve  required  is 
the  singular  solution,  which,  in  accordance  with  Art.  8,  is  the 


:24  DIFFERENTIAL   EQUATIONS. 

envelope  of  these  lines.     The  result  in  this  case  will  be  found 
to  be  the  parabola  Vx  +  Vy  =  Va. 

An  important  application  is  the  determination  of  the 
"orthogonal  trajectories"  of  a  given  system  of  curves,  that  is 
to  say,  the  curves  which  cut  at  right  angles  everyvcurve  of  the 
given  system.  The  differential  equation  of  the  trajectory  is 
readily  derived  from  that  of  the  given  system ;  for  at  every 
point  of  the  trajectory  the  value  of/  is  the  negative  reciprocal 
•of  its  value  in  the  given  differential  equation.  We  have  there- 
fore only  to  substitute  — /''  for  p  to  obtain  the  differential 
•equation  of  the  trajectory.  For  example,  let  it  be  required  to 
determine  the  orthogonal  trajectories  of  the  system  of  pa- 
rabolas 

y"^  =  4ax 

having  a  common  axis  and  vertex.     The  differential  equation 
of  the  system  found  by  eliminating  a  is 

2  xdy  =  y  dx. 

.    Putting  —  —  in  place  of  -~,  the  differential  equation  of 

the  system  of  trajectories  is 

2x  dx -\- y  dy  ■=^  Of 
whence,  integrating, 

2x''-^f  =  c\ 

The  trajectories  are  therefore  a  system  of  similar  ellipses 
with  axes  coinciding  with  the  coordinate  axes. 

Prob.  46.  Show  that  when  the  differential  equation  of  a  system 
is  of  the  second  degree,  its  discriminant  and  that  of  its  trajectory 
system  will  be  identical ;  but  if  it  represents  a  singular  solution  in 
one  system,  it  will  constitute  a  cusp  locus  of  the  other. 

Prob.  47.  Determine  the  curve  whose  subtangent  is  co,nstant  and 
equal  to  a.  Ans.  ce*=y^, 

Prob.  48.  Show  that  the  orthogonal  trajectories  of  the  curves 
r^=ic^  'sXnnB  are  the  same  system  turned  through  the  angle  —  about 
the  pole.     Examine  the  cases  n  =■  1,  n  =  2,  and  «  =  i. 

Prob.  49.   Show  that  the  orthogonal  trajectories  of  a  system  of 


SIMULTANEOUS    DIFFERENTIAL    EQUATIONS.  2'5 

circles  passing  through  two  given  points  is  another  system  of  circles 

having  a  common  radical  axis. 

ProbI  50.  Determine   the   curve   such    that   the   area   inclosed 

by  any  two  ordinates,  the  curve   and   the   axis  of  ^,  is  equal  to 

the  product  of  the    arc  and  the  constant    line  a.     Interpret    the 

singular  solution.  -      -- 

Ans.  The  catenary  ^y  =  \a{f—e  "). 

Prob.  51.  Show  that  a  system  of  confocal  conies  is  self-orthog- 
onal. Ax^^  6^j^^-At3 

Art.  12.    Simultaneous  Differential  Equations. 

A  system  of  n  equations  between  n  +  i  variables  and  their 
differentials  is  a  "determinate"  differential  system,  because  it 
serves  to  determine  the  n  ratios  of  the  differentials  ;  io  that, 
taking  any  one  of  the  variables  as  independent,  the  others  vary- 
in  a  determinate  manner,  and  may  be  regarded  as  functions  of 
the  single  independent  variable.  Denoting  the  variables  by  x^ 
y,  z,  etc.,  the  system  may  be  written  in  the  symmetrical  form 

dx  _dy  _d2  _ 

X~V~Z  * 

where  X,  Y,  Z .  ,  .  may  be  any  functions  of  the  variables. 

If  any  one  of  the  several  equations  involving  two  differen- 
tials contains  only  the  two  corresponding  variables,  it  is  an 
ordinary  differential  equation;  and  its  integral,  giving  a  re- 
lation between  these  two  variables,  may  enable  us  by  elimina- 
tion to  obtain  another  equation  containing  two  variables  only, 
and  so  on  until  n  integral  equations  have  been  obtained. 
Given,  for  example,  the  system 

dx _dy  _  dz  .  . 

X         z  ~  y'  ^  ' 

The  relation  between  dy  and  dz  above  contains  the  varia- 
bles y  and  z  only,  and  its  integral  is 

jj/^— ^''=«.  (2) 

Employing  this  to  eliminate  z  from  the  relation  between 
dx  and  dy  it  becomes 

dx  _         dy 


2t)  DIFFERENTIAL   EQUATIONS. 

of  which  the  integral  is 

y-V  V{/  +  ^)  =  bx.  (3) 

The  integral  equations  (2)  and  (3),  involving  two  constants 

of  integration,  constitute  the  complete  solution.     It  is  in  like 

manner  obvious  that  the  complete  solution  of  a  system  of  n 

equations  should  contain  n  arbitrary  constants. 

Confining  ourselves  now  to  the  case  of  three  variables,  an 
extension  of  the  geometrical  interpretation  given  in  Art.  2 
presents  itself.  Let  x,  y,  and  z  be  rectangular  coordinates  of 
P  referred  to  three  planes.  Then,  if  P  starts  from  any  given 
position  Af  the  given  system  of  equations,  determining  the 
ratios  dx\  dy\  dz,  determines  the  direction  in  space  in  which  P 
moves.*  As  P  moves,  the  ratios  of  the  differentials  (as  deter- 
mined by  the  given  equations)  will  vary,  and  if  we  suppose  P 
to  move  in  such  a  way  as  to  continue  to  satisfy  the  differential 
equations,  it  will  describe  in  general  a  curve  of  double  curva- 
ture which  will  represent  a  particular  solution.  The  complete 
solution  is  represented  by  the  system  of  lines  which  may  be 
thus  obtained  by  varying  the  position  of  the  initial  point  A. 
This  system  is  a  **  doubly  infinite  "  one  ;  for  the  two  relations 
between  x,  y,  and  z  which  define  it  analytically  must  contain 
two  arbitrary  parameters,  by  properly  determining  which  we 
can  make  the  line  pass  through  any  assumed  initial  point.^ 

Each  of  the  relations  between  x,  y  and  z,  or  integral  equa- 
tions, represents  by  itself  a  surface,  the  intersection  of  the  two 
surfaces  being  a  particular  line  of  the  doubly  infinite  system. 
An  equation  like  (2)  in  the  example  above,  which  contains  only 
one  of  the  constants  of  integration,  is  called  an  integral  of  the 
differential  system,  in  contradistinction  to  an   "  integral  equa- 

*  It  is  assumed  in  the  explanation  that  X,  V,  and  Z  are  one-valued  functions 
of  x,y,  and  z.  There  is  then  but  one  direction  in  which  F  can  move  when 
passing  a  given  point,  and  the  system  is  a  non-intersecting  system  of  lines.  But 
if  this  is  not  the  case,  as  for  example  when  one  of  the  equations  giving  the  ratio 
of  the  differentials  is  of  higher  degree  the  lines  may  form  an  intersecting  sys- 
tem, and  there  would  be  a  theory  of  singular  solutions,  into  which  we  do  not 
here  enter. 


SIMULTANEOUS   DIFFERENTIAL    EQUATIONS.  27 

tion  "  like  (3),  which  contains  both  constants.  An  integral 
represents  a  surface  which  contains  a  singly  infinite  system  of 
lines  representing  particular  solutions  selected  from  the  doubly 
infinite  system.  Thus  equation  (2)  above  gives  a  surface  on 
which  lie  all  those  lines  for  which  a  has  a  given  value,  while  b 
may  have  any  value  whatever  ;  in  other  words,  a  surface  which 
passes  through  an  infinite  number  of  the  particular  solution 
lines. 

The  integral  of  the  system  which  corresponds  to  the  con- 
stant b  might  be  found  by  eliminating  a  between  equations  (2) 
and  (3).  It  might  also  be  derived  directly  from  equation  (i)  ; 
thus  we  may  write 

dx      dy  _dz  __   dy  -\-  dz  _du 
X  ~   z  ^  y  ~~     y  -\-  ^     ~  u^ 

in  which  a  new  variable  u^^  y  -\-  zXs  introduced.  The  rela- 
tion between  dx  and  du  now  contains  but  two  variables,  and 

its  integral, 

y-^^  =  bx,  (4) 

is  the  required  integral  of  the  system  ;  and  this,  together  with 
the  integral  (2),  presents  the  solution  of  equations  (i)  in  its 
standard  form.  The  form  of  the  two  integrals  shows  that  in 
this  case  the  doubly  infinite  system  of  lines  consists  of  hyper- 
bolas, namely,  the  sections  of  the  system  of  hyperbolic  cylinders 
represented  by  (2)  made  by  the  system  of  planes   represented 

by  (4). 

A  system  of  equations  of  which  the  members  possess  a  cer- 
tain symmetry  may  sometimes  be  solved  in  the  following 
manner.     Since 

dx  _  dy  _dz  _  \dx  -|-  fxdy  -]-  vdz 

if  we  take  multipliers  A,  /i,  v  such  that 

we  shall  have  "K-dx  +  P-dy  +  '^dz  =  o. 

If  the  expression  in  the  first  member  is  an  exact  differential, 


28  DIFFERENTIAL   EQUATIONS. 

direct  integration  gives  an  integral  of  the  given  system.  For 
example,  let  the  given  equations  be 

dx       _       dy      ___        dz 
mz  —  ny~  7tx  —  lz~  ly  —  mx  * 

/,  m  and  n  form  such  a  set  of  multipliers,  and  so  also  do  x^y 
and  z.     Hence  we  have 

Idx  -\-  mdy  -\-  ndz  =  o, 

and  also  xdx  -\-  ydy-\-  zdz  =  o. 

Each  of  these  is  an  exact  equation,  and  their  integrals 

Ix  -\-  my  -^  nz  =  a 

and  x'+/  +  z'=d' 

constitute  the  complete  solution.  The  doubly  infinite  system 
of  lines  consists  in  this  case  of  circles  which  have  a  common 
axis,  namely,  the  line  passing  through  the  origin  and  whose 
direction  cosines  are  proportional  to  /,  m,  and  n. 

dx                 dy           dz 
Prob.  ^2.  Solve  the   equations  -^ ^ »=  -^^— — ,  and 

X  —  y  —  z         2xy        2XZ 

interpret  the  result  geometrically.      (Ans.  y=az^  x'^-\-y^  -\-z'^=bz.) 
dx  dy  dz 


Prob.  53.  Solve 


y  -\-  z      z  -\-  X       X  -\-  y 

Ans.  \^(x  -\-y  -{•  z)  = 


'  y        X  —  z 

Tk  i_       o  1        ^^  ^y  dz 

Prob.  54.  Solve  r, ^^ —  =  / "^ —  =  ~, r^ — • 

^^  {b  —  c)yz      {c  —  a)zx        {a  —  b)xy 

Ans.  ^'  +/  4-  2"  =  A,  ax*  4-  bf  +  cz'^  =  B. 


Art.  13.    Equations  of  the  Second  Order. 

A  relation  between  two  variables  and  the  successive  deriva- 
tives of  one  of  them  with  respect  to  the  other  as  independent 
variable  is  called  a  differential  equation  of  the  order  indicated 
by  the  highest  derivative  that  occurs.     For  example, 

is  an  equation  of  the  second  order,  in  which  x  is  the  independent 


EQUATIONS    OF    THE    SECOND    ORDER.  29 

variable.     Denoting  as  heretofore  the  iirst  derivative  by/,  this 

equation  may  be  written 

df) 
(i+^')^  +  -^/  +  ^^  =  o,  (l) 

and  this,  in  connection  with 

which  defines  /,  forms  a  pair  of  equations  of  the  first  order, 
connecting  the  variables  x,  y,  and  /.  Thus  any  equation  of  the 
second  order  is  equivalent  to  a  pair  of  simultaneous  equations 
of  the  first  order. 

When,  as  in  this  example,  the  given  equation  does  not  con- 
tain j  explicitly,  the  first  of  the  pair  of  equations  involves  only 
the  two  variables  x  and/  ;  and  it  is  further  to  be  noticed  that, 
when  the  derivatives  occur  only  in  the  first  degree,  it  is  a  linear 
equation  for/.     Integratipg  equation  (i)  as  such,  we  find 

and  then  using  this  value  of/  in  equation  (2),  its  integral  is 

y  =  c,-mx  +  c,  log  [x  +  |/(i  -I:  x')],  (4) 

in  which,  as  in  every  case  of  two  simultaneous  equations  of  the 
first  order,  we  have  introduced  two  constants  of  integration. 

An   equation   of   the  first  order  is  readily  obtained  also 

when  the  independent  variable  is  not  explicitly  contained  in 

the  equation.     The  general  equation  of  rectilinear  motion  in 

d'^s 
dynamics  affords  an  illustration.     This  equation  is  —  =  f{s)y 

where  s  denotes  the  distance  measured  from  a  fixed  center  of 

dv 
force  upon  the  line  of  motion.     It  may  be  written  -y  = /(s),  in 

dl 

ds 
connection  with  —  =  v,  which  defines  the  velocity.     Eliminat- 
ing  dt   from    these    equations,   we    have   vdv  =  f{s)ds,  whose 
integral  is  ^v^  =    I  f{s)ds  +  c,  the  "  equation  of  energy  "  for 
the  unit  mass.    The  substitution  of  the  value  found  for  v  in  the 


30  DIFFERENTIAL   EQUATIONS. 

second  equation  gives  an  equation  from  which  /  is  found  in 
terms  of  s  by  direct  integration. 

The  result  of  the  first  integration,  such  as  equation  (3)  above, 
is  called  a  "first  integral"  of  the  given  equation  of  the  second 
order ;  it  contains  one  constant  of  integration,  and  its  complete 
integral,  which  contains  a  second  constant,  is  also  the  '*  com- 
plete integral "  of  the  given  equation. 

A  differential  equation  of  the  second  order  is  "  exact  "  when, 
all  its  terms  being  transposed  to  the  first  member,  that  member 
is  the  derivative  with  respect  to  x  of  an  expression  of  the  first 
order,  that  is,  a  function  of  x,  y  and  p.  It  is  obvious  that  the 
terms  containing  the  second  derivative,  in  such  an  exact  differ- 
ential, arise  solely  from  the  differentiation  of  the  terms  con- 
taining/ in  the  fun(!tion  of  x^ y  and/.  For  example,  let  it  be 
required  to  ascertain  whether 

is  an  exact  equation.     The  terms  in  question  are  (i  —  ^"j-^-, 

which  can  arise  only  from  the  differentiation  of  {\  —  x'')p. 
Now  subtract  from  the  given  expression  the  complete  deriva- 
tive of  (i  —  x'')p,  which  is 

the  remainder  i?  x-^  -\-y,  which  is  an  exact  derivative,  namely, 
ax 

that  of  xy.  Hence  the  given  expression  is  an  exact  differ- 
ential, and 

{x-x^)^£^xy  =  c,  (6) 

is  the  first  integral  of  the  given  equation.  Solving  this  Hnear 
equation  for^,  we  find  the  complete  integral 

y^C,X-^C,S/[\     -.r').  (7) 

Prob.  55.  Solve  (i  -  x') £,  -  ^£  =  2. 

Ans.  ,)■  =  (sin~^  x^  -f-  r,  sin~^  x  +  Cy 


THE    TWO    FIRST    INTEGRALS.  31 

Prob.  56.  Solve  ^  =  f  •  Ans.  y  =  '^-\-  c,x\ 

Prob.  57.  Solve  -y^  =  a''x  —  b'^y. 

Ans.  a^x  —  b'^y  —  A  sin  bx  -\-  B  cos  bx» 
Prob.  58.  Solve  j^;0  +  (^)'  =  i.      Ans.  f  =  x^ -\-  c,x  +  c,. 


Art.  14.    The  Two  First  Integrals. 

We  have  seen  in  the  preceding  article  that  the  complete 
integral  of  an  equation  of  the  second  order  is  a  relation  be- 
tween X,  y  and  two  constants  c^  and  c^ .  Conversely,  any  rela- 
tion between  x^  y  and  two  arbitrary  constants  may  be  regarded 
as  a  primitive*  from  which  a  differential  equation  free  from  both 
arbitrary  constants  can  be  obtained.  The  process  consists  in 
first  obtaining,  as  in  Art.  3,  a  differential  equation  of  the  first 
order  independent  of  one  of  the  constants,  say  c^ ,  that  is,  a  rela- 
tion between  x,  y,p  and  <:, ,  and  then  in  like  manner  eliminating 
c^  from  the  derivative  of  this  equation.  The  result  is  the  equa- 
tion of  the  second  order  or  relation  between  x,  y,  p  and  q  {q 
denoting  the  second  derivative),  of  which  the  original  equation 
is  the  complete  primitive,  the  equation  of  the  first  order  being 
the  first  integral  in  which  c^  is  the  constant  of  integration.  It 
is  obvious  that  we  can,  in  like  manner,  obtain  from  the  primi- 
tive a  relation  between  x,  y,  p  and  c^ ,  which  will  also  be  a  first 
integral  of  the  differential  equation.  Thus,  to  a  given  form  of 
the  primitive  or  complete  integral  there  corresponds  two  first 
integrals. 

Geometrically  the  complete  integral  represents  a  doubly 
infinite  system  of  curves,  obtained  by  varying  the  values  of  c^ 
and  of  c^  independently.  If  we  regard  c^  as  fixed  and  c^  as 
arbitrary,  we  select  from  that  system  a  certain  singly  infinite 
system;  the  first  integral  containing  c^  is  the  differential  equa- 
tion of  this  system,  which,  as  explained  in  Art.  2,  is  a  relation 
between  the  coordinates  of  a  moving  point  and  the  direction 
of  its  motion  common  to  all  the  curves  of  the  system.     But 


32 


DIFFERENTIAL   EQUATIONS. 


the  equation  of  the  second  order  expresses  a  property  involv- 
ing curvature  as  well  as  direction  of  path,  and  this  property 
being  independent  of  c^  is  common  to  all  the  systems  corre- 
sponding to  different  values  of  ^„  that  is,  to  the  entire  doubly 
infinite  system.  A  moving  point,  satisfying  this  equation, 
may  have  any  position  and  move  in  any  direction,  provided  its 
path  has  the  proper  curvature  as  determined  by  the  value  of  q 
derived  from  the  equation,  when  the  selected  values  of  x,  y 
and  p  have  been  substituted  therein.* 

For  example,  equation  (7)  of  the  preceding  article  repre- 
sents an  ellipse  having  its  center  at  the  origin  and  touching 
the  lines  ;r  =  ±  i,  as  in  the  diagram  ;  c^  is  the  ordinate  of  the 
point  of  contact  with  x  •=  \^  and  c^  that  of  the  point  in  which 
the  ellipse  cuts  the  axis  oi  y.  If  we  regard  c^  as  fixed  and  c^ 
as  arbitrary,  the  equation  represents  the  system  of  ellipses 
touching  the  two  lines  at  fixed  points,  and  equation  (6)  is  the 
differential  equation  of  this  system.  In 
like  manner,  if  c^  is  fixed  and  c^  arbitrary, 
equation  (7)  represents  a  system  of  ellipses 
cutting  the  axis  of  y  in  fixed  points 
and  touching  the  lines  x  =  ±  i.  The 
corresponding  differential  equation  will  be 
found  to  be 

Finally,  the  equation  of  the  second  order,  independent  of  <:, 
and  r,  [(5)  of  the  preceding  article]  is  the  equation  of  the 
doubly  infinite  system  of  conies  f  with  center  at  the  origin, 
and  touching  the  fixed  lines  x  =  ±  i. 

*  If  the  equation  is  of  the  second  or  higher  degree  in  ^,  the  condition  for 
equal  roots  is  a  relation  between  x, y  and/,  which  may  be  found  to  satisfy  the 
given  equation.  If  it  does,  it  represents  a  system  of  singular  solutions;  each 
of  the  curves  of  this  system,  at  each  of  its  points,  not  only  touches  but  osculates 
with  a  particular  integral  curve.  It  is  to  be  remembered  that  a  singular  solu- 
tion of  a  first  integral  is  not  generally  a  solution  of  the  given  differential  equa- 
tion; for  it  represents  a  curve  which  simply  touches  but  does  not  osculate  a  set 
of  curves  belonging  to  the  doubly  infinite  system. 

f  Including  hyperbolas  corresponding  to  imaginary  values  of  ct. 


THE    TWO    FIRST    INTEGRALS.  33 

But,  starting  from  the  differential  equation  of  second  order, 
we  may  find  other  first  integrals  than  those  above  which  corre- 
spond to  c^  and  c^.  For  instance,if  equation  (5)  be  multipHed 
by/,  it  becomes 

which  is  also  an  exact  equation,  giving  the  first  integral 

in  which  c^  is  a  new  constant  of  integration. 

Whenever  two  first  integrals  have  thus  been  found  inde- 
pendently, the  elimination  of  /  between  them  gives  the  com- 
plete integral  without  further  integration.*  Thus  the  result 
of  eliminating  /  between  this  last  equation  and  the  first  inte- 
gral containing  c^  [equation  (6),  Art.  13]  is 

O  _  13       2  3  3 

/  —2c,xy  +  c,x  =c,  —  c,, 

which  is  therefore  another  form  of  the  complete  integral.  It 
is  obvious  from  the  first  integral  above  that  c^  is  the  maximum 
value  of  y,  so  that  it  is  the  differential  equation  of  the  system 
of  ellipse  inscribed  in  the  rectangle  drawn  in  the  diagram.  A 
comparison  of  the  two  forms  of  the  complete  integral  shows 
that  the  relation  between  the  constants  is  c^  =  c^  -\-  c^. 

If  a  first  integral  be  solved  for  the  constant,  that  is,  put  in 
the  form  (p{x,  y,  p)  =  c,  the  constant  will  disappear  on  differ- 
entiation, and  the  result  will  be  the  given  equation  of  second 
order  multiplied,  in  general,  by  an  integrating  factor.  We  can 
thus  find  any  number  of  integrating  factors  of  an  equation 
already  solved,  and  these  may  "Suggest  the  integrating  factors 
of  more  general  equations,  as  illustrated  in  Prob.  59  below. 

*  The  principle  of  this  method  has  already  been  applied  in  Art.  10  to  the 
solution  of  certain  equations  of  the  first  order;  the  process  consisted  of  forming 
the  equation  of  the  second  order  of  which  the  given  equation  is  a  first  integral 
(but  with  a  particular  value  of  the  constant),  then  finding  another  first  integral 
and  deriving  the  complete  integral  by  elimination  of  /. 


34  DIFFERENTIAL   EQUATIONS. 

Prob.  59.  Solve  the  equation  —^  +  ^'7  =  0  in  the  form 

y=-  A  cos  ax  -\-  B  sin  ax\ 

and  show  that  the  corresponding  integrating  factors  are  also  inte- 
grating factors  of  the  equation 

where  Jf  is  any  function  of  x\  and  thence  derive  the  integral  of  this 
equation.  . 

Ans.  ay  =  sin  ax  I  cos  ax  .  Xdx  —  cos  ax  I  sin  ax  .  Xdx. 

Prob.  60.  Find  the  rectangular  and  also  the  polar  differential 
.equation  of  all  circles  passing  through  the  origin. 

A„s.(..+y)g=.[.+(2y](.|-.).  and  .+j:=o. 


Art.  15.    Linear  Equations. 

A  linear  differential  equation  of  any  order  is  an  equation  of 
the  first  degree  with  respect  to  the  dependent  variable  y  and 
^ach  of  its  derivatives,  that  is,  an  equation  of  the  form 

where  the  coefificients  P^,  .  .  .  P„  and  the  second  member  X  are 
functions  of  the  independent  variable  only. 

The  solution  of  a  linear  equation  is  always  supposed  to  be 
in  the  form  y  =f{x) ;  and  if  y^  is  a  function  which  satisfies  the 
equation,  it  is  customary  to  speak  of  the  function  j/,,  rather  than 
of  the  equation  ^  =  ^,,  as  an  "integral"  of  the  linear  equa- 
tion. The  general  solution  of  the  linear  equation  of  the  first 
order  has  been  given  in  Art.  6.  For  orders  higher  than  the 
first  the  general  expression  for  the  integrals  cannot  be  effected 
by  means  of  the  ordinary  functional  symbols  and  the  integral 
5ign,  as  was  done  for  the  first  order  in  Art.  6. 

The  solution  of  equation  (i)  depends  upon  that  of 


LINEAR    EQUATIONS.  35 

The  complete  integral  of  this  equation  will  contain  n  arbi- 
trary constants,  and  the  mode  in  which  these  enter  the  expres- 
sion for  y  is  readily  inferred  from  the  form  of  the  equation. 
For  let  jj  be  an  integral,  and  c^  an  arbitrary  constant ;  the  re- 
sult of  putting  y  —  cj^  in  equation  (2)  is  c^  times  the  result  of 
putting  J/  =  j^, ;  that  is,  it  is  zero  ;  therefore  cj^  is  an  integral. 
So  too,  if  J2  is  an  integral,  c^y^  is  an  integral ;  and  obviously 
also  ^j/j  +  c^y^  is  an  integral.  Thus,  if  n  distinct  integrals  j^, 
y^f  .  '  yn  can  be  found, 

y  =  ^i/i  +  ^.J.  +  •   .   •  +  ^n^n  (3) 

will  satisfy  the  equation,  and,  containing,  as  it  does,  the  proper 
number  of  constants,  will  be  the  complete  integral. 

Consider  now  equation  (i);  let  F  be  a  particular  integral  "of 
it,  and  denote  by  u  the  second  member  of  equation  (3),  which 
is  the  complete  integral  when  X  =  O.     If 

y=Y+u  (4) 

be  substituted  in  equation  (i),  the  result  will  be  the  sum  of  the 
results  of  putting  y  =  Fand  of  putting  y  =  u  ;  the  first  of 
these  results  will  be  X,  because  Fis  an  integral  of  equation  (i), 
and  the  second  will  be  zero  because  u  is  an  integral  of  equa- 
tion (2).  Hence  equation  (4)  expresses  an  integral  of  (i);  and 
since  it  contains  the  n  arbitrary  constants  of  equation  (3),  it 
is  the  complete  integral  of  equation  (i).  With  reference  to 
this  equation  Fis  called  "the  particular  integral,"  and  u  is 
called  "the  complementary  function."  The  particular  integral 
contains  no  arbitrary  constant,  and  any  two  particular  integrals 
may  differ  by  any  multiple  of  a  term  belonging  to  the  comple- 
mentary function. 

If  one  term  of  the  complementary  function  of  a  linear 
equation  of  the  second  order  be  known,  the  complete  solution 
can  be  found.  For  let  j/,  be  the  known  term  ;  then,  if  y  =  y^v 
be  substituted  in  the  first  member,  the  coeflficient  of  v  in  the 
result  will  be  the  same  as  if  v  were  a  constant :  it  will  there- 
fore be  zero,  and  7^  being  absent,  the  result  will  be  a  linear  equa- 
tion of  the  first  order  for  v\  the  first  derivative  o-f  v.      Under 


3G  DIFFERENTIAL   EQUATIONS. 

the  same  circumstances  the  order  of  any  linear  equation  can 
in  like  manner  be  reduced  by  unity. 

A  very  simple  relation  exists  between  the  coefficients  of  an 
exact  linear  equation.  Taking,  for  example,  the  equation  of 
the  second  order,  and  indicating  derivatives  by  accents,  if 

is  exact,  the  first  term  of  the  integral  will  be  Pj'  Subtracting 
the  derivative  of  this  from  the  first  member,  the  remaihder  is 
(/\  —  P^)y'  +  P^y.  The  second  term  of  the  integral  must 
therefore  be  (/^^  —  P^)y ;  subtracting  the  derivative  of  this  ex- 
pression, the  remainder,  (/*,  —  P^  -f  P^")yy  must  vanish.  Hence 
/*,  —  /*/  +  P^"  =  o  is  the  criterion  for  the  exactness  of  the 
given  equation.  A  similar  result  obviously  extends  to  equa- 
tions of  higher  orders. 

d^y  dy 

Prob.  6i.  Solve  x— (3  +  x)-j-  -f  3^  =  o,  noticing  that  ^*  is 

an  integral.  Ans.  y  =  c^e^  -{-  c^(x^  +  ^x""  -f  6x  +  6. 

Prob.  62.  Solve  (x^  —  x)-^^  +  2(2^  +  i)^  +  27  =  o. 

Ans.  {x  —  lYy  =  c^{x*  —  6x^  -{-  2X  —  ^  —  4^"  log  x)  -\-  c^x^, 

Prob,  63.  Solve-j^  +  cos  6^^  —  2  sin  6-^  —y  cos  6^  =  sin  2^. 
Ans.  y^e-  s'"  ^fe «i" \cfi^  c^dB  +  c,e  -  ^'^^  -  ^'"  ^  ~  '. 


Art.  16.    Linear  Equations  with  Constant 
Coefficients. 

The  linear  equation  with  constant  coefficients  and  second 
member  zero  may  be  written  in  the  form 

A,Dy  +  A,D^-y  +  . . .  +  ^„^  =  o,  (I) 

d  <P 

in  which  D  stands  for  the  operator  -j-^  D^  for  -j-^,  etc.,  so  that 

ly  indicates  that  the  operator  is  to  be  applied  n  times.     Then, 
since  De"*""  =  me^'',  Z^V"*  =  /^V"'^  etc.,  it  is  evident   that   if 


LINEAR  EQUATIONS,  CONSTANT  COEB'FICIENTS.  *        37 

y  =  e"^^  be  substituted  in  equation  (i),  the  result  after  rejecting 
the  factor  ^'"^  will  be 

A^nf  +  A,nf-'  +  . . .  +  ^«  =  o.  (2) 

Hence,  if  m  satisfies  equation  (2),  e^''  is  an  integral  of  equation 
(i) ;  and  if  m^,  w,,  , .  .m^  are  n  distinct  roots  of  equation  (2), 
the  complete  integral  of  equation  (i)  will  be 

y  =  ^^^"'i*  +  ^^^'«a*  _|-  .  . .  -j-  c„e"''\  (3) 

For  example,  if  the  given  equation  is 
d^y       dy 

the  equation  to  determine  m  is 

rn^  —  m  —  2  =  Oy 

of  which  the  roots  are  m^  =  2,  m^=  —  i  ;  therefore  the  in- 
tegral is 

y  =  C^e'"'  +  ^a^"""" 

The  general  equation  (i)  may  be  written  in  the  symbolic 
form  /{B)  .y  =  o,  in  which  /  denotes  a  rational  integral  func- 
tion. Then  equation  (2)  is  /{m)  =  o,  and,  just  as  this  last 
equation  is  equivalent  to 

{m  —  m^){m  —  m^) . . .  {m  —  m^)  =  0,  (4) 

so  the  symbolic  equation /{D).y  =  o  may  be  written 

{D  —  m,)(D  —  ;«J  . ,  .  {D  —  m„)y  =  o.  (5) 

This  form  of  the  equation  shows  that  it  is  satisfied  by  each  of 
the  quantities  which  satisfy  the  separate  equations 

{B  —  m,)y  =  0,     (Z>  —  m^)y  =  0. .  .{D—  ni^y  =  o ;       (6) 
that  is  to  say,  by  the  separate  terms  of  the  complete  integral. 

If  two  of  the  roots  of  equation  (2)  are  equal,  say  to  m^,  two 
of  the  equations  (6)  become  identical,  and  to  obtain  the  full 
number  of  integrals  we  must  find  two  terms  corresponding  to 
the  equation 

(P-m^'y  =  0\  (7) 

in  other  words,  the  complete  integral  of  this  equation  of  which 
y^  ~  gm^x  jg  known  to  be  one  integral.     For  this  purpose  we 


38      «  DIFFERENTIAL    EQUATIONS. 

put,  as  explained  in  the  preceding  article,  j  =jjZ;.  By  differen- 
tiation, Dy  =  De*^^*v  =  ^*»*(w,7^  -f-  Dz^  ;  therefore 

{D  —  m^)e*^^*v  =  e'^^^Dv,  (8) 

In  like  manner  we  find 

(Z?  —  ^«,)V«>*z'  =  e'^^^'D^v,  (9). 

Thus  equation  (7)  is  transformed  to  D^v  =  o,  of  which  the 
complete  integral  is  z^  =  c^x  -\-c^ ;  hence  that  of  equation  (7)  is 

y  =  e-'^-{c,x+c,),  (I0> 

These  are  therefore  the  two  terms  corresponding  to  the  squared 
factor  {D  —  m^  in  f{D)y  =  o. 

It  is  evident  that,  in  a  similar  manner,  the  three  terms 
corresponding  to  a  case  of  three  equal  roots  can  be  shown  ta 
be  c^^(c^x^  +  c^x  +  ^s),  and  so  on. 

The  pair  of  terms  corresponding  to  a  pair  of  imaginary 
roots,  say  Wj  =  or  +  ^/^y  m^  =  a  —  t/3,  take  the  imaginary  form 

Separating  the  real  and  imaginary  parts  of  ^'^*  and  ^-'^*,  and 
changing  the  constants,  the  expression  becomes 

e^{A  cos  /3x-\-B  sin  /3x).  (i  i) 

For  a  multiple  pair  of  imaginary  roots  the  constants  A  and 
B  must  be  replaced  by  polynomials  as  above  shown  in  the  case 
of  real  roots. 

When  the  second  member  of  the  equation  with  constant 

coefficients  is  a  function  of  X,  the  particular  integral  can  also 

be  made  to  depend  upon  the  solution  of  linear  equations  of 

the  first  order.     In   accordance  with   the   symboHc   notation 

introduced  above,  the  solution  of  the  equation 

dy 

■^  —  ay  =  X,     or     (B-a)y  =  X  (12) 

is  denoted  by  y  =  (D  —  a)-'X,  so  that,  solving  equation  (12), 
we  have 

n^^=  ^S'-"^dx  (13) 

as  the  value  of  the  inverse  symbol  whose  meaning  is  "that 


LINEAR    EQUATIONS,  CONSTANT  COEFFICIENTS.  39^ 

function  of  x  which  is  converted  to  X  by  the  direct  operation 
expressed  by  the  symbol  D  —  a''  Taking  the  most  convenient 
special  value  of  the  indefinite  integral  in  equation  (13),  it  gives 
the  particular  integral  of  equation  (12).  In  like  manner,  the  par- 
ticular integral  oi  f{D)y  =  X  is  denoted  by  the  inverse  symbol 

-TTjf.X.     Now,  with  the  notation  employed  above,  the  symbolic 

fraction  may  be  decomposed  into  partial  fractions  with  constant 
numerators  thus : 

I  N  N  N 

lff,\^  =  n  ^+  n     '     -^  +  .  •  .  +  TT^^^^''  (H) 

/{D)  D  —  m^       ^   D  —  m^       '  ^    D  —  m„        ^  ^^ 

in  which  each  term  is  to  be  evaluated  by  equation  (13),  and 
may  be  regarded  (by  virtue  of  the  constant  involved  in  the 
indefinite  integral)  as  containing  one  term  of  the  complement- 
ary function.  For  example,  the  complete  solution  of  the 
equation 

dx'       dx        -^ 
is  thus  found  to  be 

y  =  \^-Je''^Xdx-\e-''Je^Xdx. 

When  X  is  a  power  of  x  the  particular  integral  may  be 
found  as  follows,  more  expeditiously  than  by  the  evaluation  of 
the  integrals  in  the  general  solution.  For  example,  if  X  =  ;tr* 
the  particular  integral  in  this  example  may  be  evaluated  by" 
development  of  the  inverse  symbol,  thus  : 

=  -i[i  -\D^lD^-,,  .]x^=-ix'  +  ix-l 

*  The  validity  of  this  equation  depends  upon  the  fact  that  the  operations 
expressed  in  the  second  member  of 

/{£>)  =  (D  -  m,\D  -m^)-^..,-^{D-mn^ 
are  commutative,  hence  the  process  of  verification  is  the  same  as  if  the  equation 
were  an  algebraic  identity.  This  general  solution  was  published  by  Boole  in 
the  Cambridge  Math.  Journal,  First  Series,  vol.  ii,  p.  114.  It  had,  however, 
been  previously  published  by  Lobatto,  Th^orie  des  Charact^ristiques,  Amster- 
dam, 1837. 


40  DIFFERENTIAL   EQUATIONS. 

The  form  of  the  operand  shows  that,  in  this  case,  it  is  only 
necessary  to  carry  the  development  as  far  as  the  term  contain- 
ing D\ 

For  other  symbolic  methods  applicable  to  special  forms  of 
X  we  must  refer  to  the  standard  treatises  on  this  subject. 

Prob.  64.  Solve  4^  -  3-^  +^  =  o. 

Ans.  y  —  ^''(Ax  -\- B)  •\-  ce'\ 
Prob.  65.  Show  that     Tr^^''  =  -^\^ 

and  that     77^  sin  (a^  +  ^)  =  iv^^^r  sin  («a:  + /5). 

Prob.  (id.  Solve  (Z>'  -{-  i  )7  =  ^  +  sin  2x -\-  sin  x,  (Compare 
Prob.  59,  Art.  14.) 

Ans.  y^  A  s\n  X  -\-  B  cos  x-\-\e^  —  \^\n  2x  —  \x  cos  x. 

Art.  17.    Homogeneous  Linear  Equations. 
The  linear  differential  equation 

^•"'"S + ^■''""£^ + . . . + ^> = o.     (I) 

in  which  A^,  A^,  etc.,  are  constants,  is  called  th'e  "homogene- 
ous linear  equation."  It  bears  the  same  relation  to  x"^  that 
the  equation  with  constant  coefficients  does  to  ^*"*.  Thus,  if 
j=;r"*  be  substituted  in  this  equation,  the  factor  x""  will  divide 
out  from  the  result,  giving  an  equation  for  determining  m^ 
and  the  n  roots  of  this  equation  will  in  general  determine  the 
n  terms  of  the  complete  integral.  For  example,  if  in  the 
equation 

we  put  y  =  x^y  the  result  is  m{m  —  i)  +  2m  —  2  =  0,  or 
{m  —  i)(7n  +  2)  =  o. 

The  roots  of  this  equation  are  m^=  i  and  w,  =  —  2. 
Hence  y  z=  c^x  -\-  c^x~* 

is  the  complete  integral. 

Equation  (i)  might  in  fact  have  been  reduced  to  the  form 
with  constant  coefficients  by  changing  the  independent  vari- 


HOMOGENEOUS   LINEAR    EQUATIONS.  41 

able  to  6,  where  x  =1  e^,  or  6  =  log  x.  We  may  therefore  at 
once  infer  from  the  results  established  in  the  preceding  article 
that  the  terms  corresponding  to  a  pair  of  equal  roots  are  of  the 
form 

(^1  +  ^,  log  ^y^    .  (2) 

and  also  that  the  terms  corresponding  to  a  pair  of  imaginary- 
roots,  a  ±  z/3,  are 

x^lA  cos  (/?  log  ^^)-\-  B  sin  {j3  log  x)].  (3) 

The  analogy  between  the  two  classes  of  linear  equations 
considered  in  this  and  the  preceding  article  is  more  clearly 
seen  when  a  single  symbol  i&=  xD  is  used  for  the  operation  of 
taking  the  derivative  and  then  multiplying  by  x,  so  that 
■&x*^  =  mx"".  It  is  to  be  noticed  that  the  operation  x'^D^  is  not 
the  same  as  i&"  or  xDxD,  because  the  operations  of  taking  the 
derivative  and  multiplying  by  a  variable  are  not  ''commu- 
tative," that  is,  their  order  is  not  indifferent.  We  have,  on  the 
contrary,  x^V  =  ■&{■&  —  i) ;  then  the  equation  given  above, 
which  is 

{x'D' +  2xB  —  2)j/ =  o, 
becomes 

[d{d  —  i)  +  2^  —  2] J/  =  o,     or     (i&  -  i)(^  +  2>  =  o, 

the  function  of  ■&  produced  being  the  same  as  the  function  of 
m  which  is  equated  to  o  in  finding  the  values  of  m. 

A  linear  equation  of  which  the  first  member  is  homoge- 
neous and  the  second  member  a  function  of  x  may  be  reduced 
to  the  form 

y(«)./  =  x.  (4) 

The  particular  integral  may,  as  in  the  preceding  article  (see 
eq.  (14)),  be  separated  into  parts  each  of  which  depends  upon 
the  solution  of  a  linear  equation  of  the  first  order.  Thus, 
solving  the  equation 

x±-af  =  X,     or     {&-a)y  =  X,  (5) 


we  find 


--X=x^  fx-^-'Xdx.  (6) 


•0- 
The  more  expeditious  method   which   may  be   employed 


43  DIFFERENTIAL   EQUATIONS, 

when  -AT is  a  power  of  x  is  illustrated  in  the  following  example : 

Given  x*  -4  —  2-f-  =  x"".    The  first  member  becomes  homo- 
dx         ax 

geneous  when  multiplied  by  x^  and  the  reduced  equation  is 

The  roots  oi  f{^)  =  o  are  3  and  the  double  root  zero,  hence 
the  complementary  function  is  ^,^' +  ^a  +  ^s  log  x.  Since  in 
general /(i^);!:*"  = /(r);r'',  we  infer  that  in  operating  upon  x^  we 
may  put  i&  =  3.     This  gives  for  the  particular  integral 


0-3^'  9^-3' 

but  fails  with  respect  to  the  factor  ^  —  3.*     We  therefore 
now  fall  back  upon  equation  (6),  which  gives 

_ x^  =  x^  I  x~'dx  =  x^  log  X. 

^-3  *^ 

The  complete  integral  therefore  is 

y  =  c,x'  +  c^  +  c,  log  X  +  ix*  log  X. 

Prob.  67.  Solve  2^"^  +  S^j-  —  3^'  =  ^'• 

Ans.  y  =  c^x-\-  c^x'l  4-  \x*. 

Prob.  68.  Solve  {x'^D*  +  3^Z>'  +  jy)y  =  -. 

Ans.  y  =  c,-^c^\ogx-\-  cX\og  xY  +  J(log  x)\ 

Art.  18.    Solutions  in  Infinite  Series. 

We  proceed  in  this  article  to  illustrate  the  method  by 
which  the  integrals  of  a  linear  equation  whose  coefificients  are 
algebraic  functions  of  x  may  be  developed  in  series  whose 
terms  are  powers  of  x.  For  this  purpose  let  us  take  the 
equation 

*  The  failure  occurs  because  x'^  is  a  term  of  the  complementary  function 
having  an  indeterminate  coefficient;  accordingly  the  new  term  is  of  the  same 
form  as  the  second  term  necessary  when  3  is  a  double  root,  but  of  course  with 
a  determinate  coefficient. 


SOLUTIONS    IN    INFINITE    SERIES.  43» 

which  is  known  as  "  Bessel's  Equation,**  and  serves  to  define^ 
the  "Besselian  Functions." 

If  in  the  first  member  of  this  equation  we  substitute  for^ 
the  single  term  Ax"'  the  result  is 

A{m'  -  n')x^  +  Ax^^\  (2) 

the  first  term  coming  from  the  homogeneous  terms  of  the 
equation  and  the  second  from  the  term  x^y  which  is  of  higher 
degree.  If  this  last  term  did  not  exist  the  equation  would  be 
satisfied  by  the  assumed  value  of/,  if  mwcrc  determined  so  as 
to  make  the  first  term  vanish,  that  is,  in  this  case,  by  Ax""  or 
Bx'"*.  Now  these  are  the  first  terms  of  two  series  each  of 
which  satisfies  the  equation.  For,  if  we  add  to  the  value  of  jr 
a  term  containing  x*''+^,  thus/  =  A^^x*"  +  A^x"''^^,  the  new  term, 
will  give  rise,  in  the  result  of  substitution,  to  terms  containing. 
^m+2  ^^^  ^m+4  respectively,  and  it  will  be  possible  so  to  take 
A^  that  the  entire  coefficient  of  x"*"^^  shall  vanish.  In  like 
manner  the  proper  determination  of  a  third  term  makes  the 
coefficient  of  ;ir"*+'*  in  the  result  of  substitution  vanish,  and  so 
on.     We  therefore  at  once  assume 

/  =  1  y^^'«+^'-  =  A^x""  +  A, x"'+'  +  A^x'^^^  +  .  .  .  ,       (3) 

in  which  r  has  all  integral  values  from  o  to  00 .  Substituting 
in  equation  (i) 

2[{{m  +  2ry—  n']A^'^+'''+  A^^+'^''+'^]  =  o.  (4) 

0 

The  coefficient  of  each  power  of  x  in  this  equation  must  sep- 
arately vanish  ;  hence,  taking  the  coefficient  of  x'^+^^'y  we  have 

[{m  +  2ry~n^]A^+A^.,=o,  (5) 

When  r  =  o,  this  reduces  to  m^  —  n^  =  0,  which  determines 
the  values  of  m,  and  for  other  values  of  r  it  gives 

*"  ~  ~  (m  -\-2r-\-  n){m  -\-  2r  —  n)    '"^*  ^  ^ 

the  relation  between  any  two  successive  coefficients. 

For  the  first  value  of  m,  namely  ;?,  this  relation  becomes     ^' 

A  I  4     . 


44  DIFFERENTIAL   EQUATIONS. 

whence,  determining  the  successive  coefficients  in  equation  (3), 
the  first  integral  of  the  equation  is 

Ao'.  =  A.x'[i  -  ;^  7,  +  („+ii„  +  2)  aTil  -  •  •  •]•  ^7) 
In  like  manner,  the  other  integral  is  found  to  be 

g./.  =  ^o^-[i  +  ;r^g+(„_,;(„_,)^,+...].(8) 

and  the  complete  integral  is_^  =  A^^  -\-  Bj^^.* 

This  example  illustrates  a  special  case  which  may  arise  in 
this  form  of  solution.  If  «  is  a  positive  integer,  the  second 
series  will  contain  infinite  coefficients.  For  example,  if  n  =  2, 
the  third  coefficient,  or  B^j  is  infinite,  unless  we  take  B„  =  o,  in 
which  case  B,  is  indeterminate  and  we  have  a  repetition  of  the 
solution  jj.  This  will  always  occur  when  the  same  powers  of 
j;  occur  in  the  two  series,  including,  of  course,  the  case  in  which 
m  has  equal  roots.  For  the  mode  of  obtaining  a  new  integral 
in  such  cases  the  complete  treatises  must  be  referred  to.f 

It  will  be  noticed  that  the  simplicity  of  the  relation  between 
consecutive  coefficients  in  this  example  is  due  to  the  fact  that 
equation  (i)  contained  but  two  groups  of  terms  producing 
different  powers  of  x,  when  Ax"'  is  substituted  for  j^  as  in  ex- 
pression (2).  The  group  containing  the  second  derivative 
necessarily  gives  rise  to  a  coefficient  of  the  second  degree  in 
m,  and  from  it  we  obtained  two  values  of  m.  Moreover,  be- 
cause the  other  group  was  of  a  degree  higher  by  two  units,  the 
assumed  series  was  an  ascending  one,  proceeding  by  powers 
oi  x\ 

*  The  Besselian  function  of  the  «th  order  usually  denoted  by  /n  is  the  value 

.  of,;;/)  above,  divided  by  2««!  if  w  is  a  positive  integer,  or  generally  by  2'*r(«+i). 

For  a  complete  discussion   of  these  functions  see  Lommel's  Studien  iiber  die 

Bessel'schen    Functionen,   Leipzig,  1868;    Todhunter's    Treatise  on  Laplace's, 

Lamp's  and  Bessel's  Functions,  London,  1875,  etc. 

f  A  solution  of  the  kind  referred  to  contains  as  one  term  the  product  of  the 
regular  solution  and  log  ;«■,  and  is  sometimes  called  a  "  logarithmic  solution." 
See  also  American  Journal  of  Mathematics,  Vol.  XI,  p.  37.  In  the  .case  of 
Bessel's  equation,  the  logarithmic  solution  is  the  "Besselian  Function  of  the 
second  kind." 


SOLUTIONS    IN    INFINITE   SERIES.  45"' 

In  the  following  example, 

d^'y    ,     dy  y 

there  are  also  two  such  groups  of  terms,  and  their  difference 
of  degree  shows  that  the  series  must  ascend  by  simple  powers. 
We  assume  therefore  at  once 

y  =  i,  A^x^-^.  (I0> 

The  result  of  substitution  is 

^[\(m+r){m-\-r- 1)-2] A^"'+'-'  +  a{m+r)A^'^-^-'']=  o.  (li> 

Equating  to  zero  the  coefficient  of  x'^'^*"^^ 

{m-{-r-\-  \){in  +  r  —  2)A^  +  a(m  +  r  —  i)A^_^  =  o,  (12) 
which,  when  r  =0,  gives 

{in  +  i){m  -2)A,=  o,  (13) 

and  when  r  >  o, 

_  m-\-r  —  I 

The  roots  of  equation  (13)  are  w  =  2  and  /«  =  —  I;  taking 
w  =  2,  the  relation  (14)  becomes 

vhc  nee  the  first  integral  is 
-4.^.  =  ^.^{l  -J<..  +  - ^.V-^^^V +...].  (15) 

Taking  the  second  value  m  =  —  i,  equation  (14)  gives 

__  r  —  2 

Br  —  —  a,    _  i)     '^  * 

whence  B^  =  —  -B^ ,  and  ^,  =  o  *;  therefore  the  second  inte* 
gral  is  the  finite  expression 

5.^,  =  i?..-[i-i..]=4i-f].  (,6) 

*^3  would  take  the  indeterminate  form,  and  if  we  suppose  it  to  have  a  finite 
value,  the  rest  of  the  series  is  equivalent  to  Bzy\,  reproducing  the  first  integral. 


46  DIFFERENTIAL    EQUATIONS. 

When  the  coefficient  of  the  term  of  highest  degree  in  the 
.result  of  substitution,  such  as  equation  (ii),  contains  w,  it  is 
•possible  to  obtain  a  solution  in  descending  powers  of  x.  In 
this  case,  m  occurring  only  in  the  first  degree,  but  one  such 
solution  can  be  found ;  it  would  be  identical  with  the  finite 
integral  (i6).  In  the  general  case  there  will  be  two  such  solu- 
tions, and  they  will  be  convergent  for  values  of  x  greater  than 
lunity,  while  the  ascending  series  will  converge  for  values  less 
than  unity.* 

When  the  second  member  of  the  equation  is  a  power  of  x^ 
the  particular  integral  can  be  determined  in  the  form  of  a  series 
in  a  similar  manner.  For  example,  suppose  the  second  mem- 
ber of  equation  (9)  to  have  been  x^.  Then,  making  the  sub- 
stitution as  before,  we  have  the  same  relation  between  consecu- 
tive coefficients;  but  when  r  =  o,  instead  of  equation  (13)  we 

have 

{m  +  \){ni  —  2)A,x"'-'  =  x^ 

to  determine  the  initial  term  of  the  series.  This  gives  m  =  2^ 
and  ^0  =  4^;  hence,  putting  m  =  ^  in  equation  (14),  we  find  for 
the  particular  integral  f 

y  =  ixi[i  -  ?^ax  +  A^-l-a'x^  -  .  .  .1 
71-        9-3  9-II-3.5  -1 

A  linear  equation  remains  linear  for  two  important  classes 

of   transformations ;    first,  when    the   independent  variable   is 

changed  to  any  function  of  x,  and  second,  when  for  y  we  put 

vf{x).     As  an  example  of  the  latter,  let  y  =  e'^'^'v  be  substituted 

in   equation  (9)  above.     After   rejecting  the   factor  e~'^,  the 

result  is 

d?V         dv       2v 

—  —  a  — =0. 

dx^         dx       x"^ 

Since  this  differs  from  the  given  equation  only  in  the  sign 

*When  there  are  two  groups  of  terms,  the  integrals  are  expressible  in  terms 
of  Gauss's  "  Hypergeometric  Series." 

f  If  the  second  member  is  a  term  of  the  complementary  function  (for  ex- 
ample, in  this  case,  if  it  is  any  integral  power  of  x),  the  particular  integral  will 
take  the  logarithmic  form  referred  to  in  the  foot-note  on  p.  346. 


SYSTEMS  OF   DIFFERENTIAL   EQUATIONS.  47 

of  a,  we  infer  from  equation  (i6)  that  it  has  the  finite  integral 

V  ^  -A — .     Hence  the  complete  integral  of  equation  (9)  can 
X       2 

be  written  in  the  form 

;rj/  =  ^,(2  —  ax)  -\-  ^,^"'"'(2  +  (^x), 

Prob.  69.  Integrate  in  series  the  equation  -j-^  -\-  xy  ^=-  o. 

Prob.  70.   Integrate  in  series  oc"—^  +  ^'-f  +  (-^  —  2)^  =  o. 

Prob.    71.    Derive    for   the    equation    of   Prob.   70  the  integral 
y^  =  e"''{x~^  -{-  1  -\-  ^x)j  an.d  find  its  relation  to  those  found  above. 


Art.  19.    Systems  of  Differential  Equations. 

It  is  shown  in  Art.  12  that  a  determinate  system  of  n  differ- 
ential equations  of  the  first  order  connecting  n  -\-  i  variables 
has  for  its  complete  solution  as  many  integral  equations  con- 
necting the  variables  and  also  involving  n  constants  of  inte- 
gration. The  result  of  eliminating  n  —  i  variables  would  be  a 
single  relation  between  the  rerhaining  two  variables  containing 
in  general  the  n  constants.  But  the  elimination  may  also  be 
effected  in  the  differential  system,  the  result  being  in  general 
an  equation  of  the  nth.  order  of  which  the  equation  just  men- 
tioned is  the  complete  integral.  For  example,  if  there  were 
two  equations  of  the  first  order  connecting  the  variables  x  and 
y  with  the  independent  variable  /,  by  differentiating  each  we 
should  have  four  equations  from  which  to  eliminate  one  vari- 
able,  say  J,  and  its  two  derivatives*  with  respect  to  /,  leaving 
a  single  equation  of  the  second  order  between  x  and  /. 

It  is  easy  to  see  that  the  same  conclusions  hold  if  some  of 
the  given  equations  are  of  higher  order,  except  that  the  order 
of  the  result  will  be  correspondingly  higher,  its  index  being  in 

*  In  general,  there  would  be  n^  equations  from  which  to  eliminate  «  —  i 
variables  and  n  derivatives  of  each,  that  is,  {n  —  i){n  +  i)  =  «'  —  i  quantities 
leaving  a  single  equation  of  the  wth  order. 


J  = 


2/         —D 

o     4^-3 


48  DIFFERENTIAL  EQUATIONS. 

general  the  sum  of  the  indices  of  the  orders  of  the  given  equa- 
tions. The  method  is  particularly  applicable  to  linear  equa- 
tions with  constant  coefficients,  since  we  have  a  general  method 
of  solution  for  the  final  result.  Using  the  symbolic  notation, 
the  differentiations  are  performed  simply  by  multiplying  by 
the  symbol  D,  and  therefore  the  whole  elimination  is  of  exactly 
the  same  form  as  if  the  equations  were  algebraic.^  For  ex- 
ample, the  system 

tPy      dx  _,  dx  ,     dy 

when  written  symboHcally,  is 

(2Z>'  -  4)y  -  Dx  —  2/,        2Dy-\-{i\D -  i)x  =  o; 

whence,  eliminating  x, 

2D' -4       -D 

2D        4D-S 

which  reduces  to 

(D-i)X2£>  +  3)y  =  2-it. 
Integrating, 

the  particular  integral  being  found  by  symbolic  development, 
as  explained  at  the  end  of  Art.  i6. 

The  value  of  x  found  in  like  manner  is 

x  =  {A'  +  B'ty+re-^-i. 

The  complementary  function,  depending  solely  upon  the  deter- 
minant  of  the  first  members,"^  is  necessarily  of  the  same  form 
as  that  for  y,  but  involves  a  new  set  of  constants.  The  re- 
lations between  the  constants  is  found  by  substituting  the 
values  of  x  and^  in  one  of  the  given  equations,  and  equating 
to  zero  in  the  resulting  identity  the  coefficients  of  the  several 
terms  of  the  complementary  function.  In  the  present  ex- 
ample we  should  thus  find  the  value  of  x,  in  terms  of  A,  B^ 
and  C,  to  be 

x  =  {6B-2A-  2Bty  -  \Ce-^'  -  f 

*  The  index  of  the  degree  in  D  of  this  determinant  is  that  of  the  order  of 
the  final  equation  ;  it  is  not  necessarily  the  sum  of  the  indices  of  the  orders  of 
the  given  equations,  but  cannot  exceed  this  sum. 


SYSTEMS   OF    DIFFERENTIAL   EQUATIONS.  49^ 

In  general,  the  solution  of  a  system  of  differential  equations 
depends  upon  our  ability  to  combine  them  in  such  a  way  as 
to  form  exact  equations.  For  example,  from  the  dynamical 
system 

-df-^'      le-^'      df-^'  ^'> 

where  X,  V,  Z  are  functions  of  x,  y,  and  ^,  but  not  of  t^ 
we  form  the  equation 

dx  jdx  ,  dy  ,dz   ,  dz  .dz       „  ,     ,    ^^^     ,  ■     , 
—  d—r  +  -~d-r-\-  —d—  —  Xdx  +  Ydy  +  Zds, 
dt    dt^ dt    dt    ^   dt    dt  '        "^   ' 

The  first  member  is  an  exact  differential,  and  we  know  that  for 
a  conservative  field  of  force  the  second  member  is  also  exact,, 
that  is,  it  is  the  differential  of  a  function  U  of  Xy  y^  and  z. 
The  integral 

is  that  first  integral  of  the  system  (i)  which  is  known  as  the 
equation  of  energy  for  the  unit  mass. 

Just  as  in  Art.  13  an  equation  of  the  second  order  was  re- 
garded as  equivalent  to  two  equations  of  the  first  order,  so  the 
system  (i)  in  connection  with  the  equation  defining  the  resolved 
velocities  forms  a  system  of  six  equations  of  the  first  order,  of 
which  system  equation  (2)  is  an  "  integral  "  in  the  sense  ex- 
plained in  Art.  12. 

Prob.  72.  Solve  the  equations =  —  ^  dt  a.^  3.  system  tm- 

—  my       mx  ^ 

ear  in  /.       Ans.  x  =  A  cos mt-\-B  sin m/fy=A  smmt—B  cosmt. 
Prob.  73.  Solve  the  system \-  n^y  =  ^,  -3^  -j-  5  =  o. 

Ans.  y  =  ^,?«^  +  Be-''''  +  -f—,  z  =  -  nAe^*-\-  tiBe  "  ***  -  -^^. 
n  —  *  «  —  I 

d^x  d^v 

Prob.  74.  Find  for  the  system  -jt  —  x(t>{x,y),  —7-  —  ycf){xyy) 

a  first  integral  independent  of  the  function  0. 


.  dy         dx        „ 

Ans.  X-—  —  y—-  =  C. 

d^       ^  dt 


50  DIFFERENTIAL   EQUATIONS. 

Prob.  75.  The  approximate  equations  for  the  horizontal  motion 
of  a  pendulum,  when  the  earth's  rotation  is  taken  into  account,  are 

show  that  both  x  and  y  are  of  the  form 

A  cos  «,/  +  ^  sin  «,/  +  C  cos  n^t  +  Z>  sin  «,/. 

Art.  20.    First  Order  and  Degree  with  Three 
Variables. 

The  equation  of  the  first  order  and  degree  between  three 
variables  Xy  y  and  z  may  be  written 

Pdx+Qdy-\-Rdz  =  Q,  (i) 

where  -P,  Q  and  R  are  functions  of  x,  y  and  z.  When  this 
equation  is  exact,  P,  Q  and  R  are  the  partial  derivatives  of 
some  function  u,  of  ;r,  y  and  z ;  and  we  derive,  as  in  Art.  4, 

-dP^-dQ      ^^a^       dR^dP^  , 

dy     dx'     dz      dy*     dx     d^  ^^^ 

for  the  conditions  of  exactness.  In  the  case  of  two  variables, 
when  the  equation  is  not  exact  integrating  factors  always  exist; 
but  in  this  case,  there  is  not  always  a  factor  /i  such  that  /aPj 
fxQ  and  ^R  (put  in  place  of  P,  Q,  and  R)  will  satisfy  all  three 
of  the  conditions  (2).  It  is  easily  shown  that  for  this  purpose 
the  relation 

\dz      dyf^     \dx     dz)^     \dy      dxJ  ^^^ 

must  exist  between  the  given  values  of  P,  Q,  and  R.  This  is 
therefore  the  "  condition  of  integrability  "  of  equation  (i).* 

When  this  condition  is  fulfilled  equation  (i)  may  be  inte- 
grated by  first  supposing  one  variable,  say  Zy.  to  be  constant. 
Thus,  integrating  Pdx  +  Qdy  =  o,  and  supposing  the  constant 
of  integration  dT  to  be  a  function  of  z,  we  obtain  the  integral,  so 

*When  there  are  more  than  three  variables  such  a  condition  of  integra- 
bility exists  for  each  group  of  three  variables,  but  these  conditions  are  not  all 
independent.  Thus  with  four  variables  there  are  but  three  independent  con* 
ditions. 


FIRST  ORDER  AND  DEGREE,  THREE  VARIABLES.  ^l 

far  as  it  depends  upon  x  and  f.  Finally,  by  comparing  the 
total  differential  of  this  result  with  the  given  equation  we  de- 
termine dC  in  terms  of  s'  and  ds,  and  thence  by  integration  the 
value  of  C. 

It  may  be  noticed  that  when  certain  terms  of  an  exact 
equation  forms  an  exact  differential,  the  remaining  terms  must 
also  be  exact.  It  follows  that  if  one  of  the  variables,  say  2 
can  be  completely  separated  from  the  other  two  (so  that  in 
equation  (i)  R  becomes  a  function  of  2  only  and  P  and  Q  func- 
tions of  X  and  y^  but  not  of  z)  the  terms  Pdx  -\-  Qdy  must  be 
thus  rendered  exact  if  the  equation  is  integrable.*  For  example, 

zydx  —  zxdy  —  y'^dz  =  o. 
is  an  integrable  equation.    Accordingly,  dividing  by  y^z,  which 
we  notice  separates  the  variable  z  from  x  and  y,  puts  it  in  the 

exact  form 

ydx  —  xdy      dz 


of  which  the  integral  is  ;i;  =^  log  cz. 

Regarding  x,  y  and  z  as  coordinates  of  a  moving  point, 
an  integrable  equation  restricts  the  point  to  motion  upon  one 
of  the  surfaces  belonging  to  the  system  of  surfaces  represented 
by  the  integral ;  in  other  words,  the  point  {x,  y,  z)  moves  in  an 
arbitrary  curve  drawn  on  such  a  surface.  Let  us  now  consider 
in  what  way  equation  (i)  restricts  the  motion  of  a  point  when 
it  is  not  integrable.  The  direction  cosines  of  a  moving  point 
are  proportional  to  dx,  dy,  and  dz\  hence,  denoting  them  by 
/,  m  and  n,  the  direction  of  motion  of  the  point  satisfying 
■equation  (1)  must  satisfy  the  condition 

P/+  Qm-\-  Rn  =  Q.  (4) 

It  is  convenient  to  consider  in  this  connection  an  auxiliary 
system  of  lines  represented,  as  explained  in  Art.  12,  by  the 
simultaneous  equations 

dx  _dy  _dz 

T~~Q~~R'  (5) 

*  In  fact  for  this  case  the  condition  (3)  reduces  to  its  last  terra,  which  ex« 
presses  the  exactness  of  Pdx  -\-  Qdy. 


52  DIFFERENTIAL   EQUATIONS. 

The  direction  cosines  of  a  point  moving  in  one  of  the  lines 
of  this  system  are  proportional  to  Py  Q  and  R.  Hence,  de- 
noting them  by  A,  //,  v,  equation  (4)  gives 

A/  -|-  ixm  -{-  vn  =  o  (6) 

for  the  relation  between  the  directions  of  two  moving  points, 
whose  paths  intersect,  subject  respectively  to  equation  (i)  and 
to  equations  (5).  The  paths  in  question  therefore  intersect  at 
right  angles;  therefore  equation  (i)  simply  restricts  a  point  ta 
move  in  a  path  which  cuts  orthogonally  the  lines  of  the  auxili- 
ary system.  ^ 

Now,  if  there  be  a  system  of  surfaces  which  cut  the  auxiliary 
lines  orthogonally,  the  restriction  just  mentioned  is  completely 
expressed  by  the  requirement  that  the  line  shall  lie  on  one  of 
these  surfaces,  the  line  being  otherwise  entirely  arbitrary* 
This  is  the  case  in  which  equation  (i)  is  integrable.* 

On  the  other  hand,  when  the  equation  is  not  integrable,  the 
restriction  can  only  be  expressed  by  two  equations  involving 
an  arbitrary  function.  Thus  if  we  assume  in  advance  one  such 
relation,  we  know  from  Art.  12  that  the  given  equation  (i) 
together  with  the  first  derivative  of  the  assumed  relation  forms 
a  system  admitting  of  solution  in  the  form  of  two  integrals* 
Both  of  these  integrals  will  involve  the  assumed  function.  For 
any  particular  value  of  that  function  we  have  a  system  of  lines 
satisfying  equation  (i),  and  the  arbitrary  character  of  the  func- 
tion makes  the  solution  sufficiently  general  to  include  all  lines 
which  satisfy  the  equation.f 

Prob.  76.  Show  that  the  equation 

{mz  —  ny)dx  +  {nx  —  lz)dy  -\-  {ly  —  mx)dz  =  0  ^ 

is  integrable,  and  infer  from  the  integral  the  character  of  the  auxil- 

*  It  follows  that,  with  respect  to  the  system  of  lines  represented  by  equations; 
(5),  equation  (3)  is  the  condition  that  the  system  shall  admit  of  surfaces  cutting 
them  orthogonally.  The  lines  of  force  in  any  field  of  conservative  forces  form 
such  a  system,  the  orthogonal  surfaces  being  the  equipotential  surfaces, 

f  So  too  there  is  an  arbitrary  element  about  the  path  of  a  point  when  the 
single  equation  to  which  it  is  subject  is  integrable,  but  this  enters  only  into  one 
of  the  two  equations  necessary  to  define  the  path. 


PARTIAL    EQUATIONS,    FIRST    ORDER.  Ho 

iary  lines.     (Compare  the  illustrative  example  at  the  end  of  Art.  12.) 

Ans.  nx  —  /z  =  C(ny  —  mz). 
Prob.  77.  Solve 7^Vji£:  —  z^dy  —  e^dz  =  o.      Ans.  yz  =^"^(1+^5?). 
Prob.  78.  Find  the  equation  which  in  connection  with^*  —.  f{^x) 
forms  the  solution  of  dz  =  aydx  +  bdy. 

Prob.  79.  Show  that  a  general  solution  of 
ydx  =  (^  —  z){dy  —  dz) 
is  given  by  the  equations 

y  —  Z—  (p{x)^  y={x  —  z)(f)\x), 

(This  is  an  example  of  "  Monge's  Solution.") 


Art.  21.    Partial  Differential  Equations  of  First 
Order  and  Degree. 

Let  z  denote  an  unknown  function  of  the  two  independent 
variables  x  and  y^  and  let 

denote  its  partial  derivatives :  a  relation  between  one  or  both 
•of  these  derivatives  and  the  variables  is  called  a  "  partial  dif- 
ferential equation  "  of  the  first  orderl  A  value  of  ^  in  terms  of 
X  and  y  which  with  its  derivatives  satisfies  the  equation,  or  a 
relation  between  x,  y  and  2  which  makes  z  implicitly  such  a 
function,  is  a  "  particular  integral."  The  most  general  equation 
of  this  kind  is  called  the  *'  general  integral." 

If  only  one  of  the  derivatives,  say/,  occurs,  the  equation 
may  be  solved  as  an  ordinary  differential  equation.  For  if  j^  is 
considered  as  a  constant,/  becomes  the  ordinary  derivative  of 
-S"  with  respect  to  ;ir ;  therefore,  if  in  the  complete  integral  of 
the  equation  thus  regarded  we.  replace  the  constant  of  integra- 
tion by  an  arbitrary  function  of  y,  we  shall  have  a  relation 
which  includes  all  particular  integrals  and  has  the  greatest  pos- 
sible generality.  It  will  be  found  that,  in  like  manner,  when 
both  /  and  q  are  present,  the  general  integral  involves  an  arbi- 
trary function. 

We  proceed  to  g,\v&  Lagrange's  solution   of  the  equation  of 


64  DIFFERENTIAL    EQUATIONS. 

the  first  order  and  degree,  or  **  linear  equation,"  which  may  be 
written  in  the  form 

Pp+Qq  =  R,  (l> 

P,  Q  and  R  denoting  functions  of  x,  y  and  z.  Let  u  =  a,  in 
which  «  is  a  function  of  x,  y  and  z,  and  a,  a  constant,  be  an 
integral  of  equation  (i).  Taking  derivatives  with  respect  to  x 
and  y  respectively,  we  have 

and  substitution  of  the  values  of  p  and  q  in  equation  (i)  gives 
the  symmetrical  relation 

Consider  now  the  system  of  simultaneous  ordinary  differ- 
ential equations 

dx       dy      dz 

P~  Q~  R  ^^^ 

Let  «  =  ^  be  one  of  the  integrals  (see  Art.  12)  of  this  sys- 
tem.    Taking  its  total  differential, 

■—-dx  +  :r-dy  +  tt-^z  =  o ; 
^x       ^  dy  dz  * 

and  since  by  equations  (3)  dx,  dy  and  dz  are  proportional  to  P^ 
Q  and  R,  we  obtain  by  substitution 

dx^+dy^-^d^^-""' 
which  is  identical  with   equation   (2).     It   follows  that  every 
integral  of  the  system  (3)  satisfies  equation  (i),  and  conversely, 
so  that  the  general  expression  for  the  integrals  of  (3)  will  be 
the  general  integral  of  equation  (i). 

Now  let  z;  =  ^  be  another  integral  of  equations  (3),  so  that 
V  is  also  a  function  which  satisfies  equation  (2).  As  explained 
in  Art.  12,  each  of  the  equations  u  =  a,  v  =  b  is  the  equation 
of  a  surface  passing  through  a  singly  infinite  system  of  lines 
belonging  to  the  doubly  infinite  system  represented  by  equa- 
tions (3).     What  we  require  is  the  general  expression  for  any 


PARTIAL   EQUATIONS,    FIRST    ORDER. 


55 


surface  passing  through  lines  of  the  system  (and  intersecting 
none  of  them).  It  is  evident  that  f{u^  v)  =f{aj  b)  =^  C  is  such 
an  equation,*  and  accordingly /"(?^,  v)^  where  f  is  an  arbitrary 
function,  will  be  found  to  satisfy  equation  (2).  Therefore,  to 
solve  equation  (i),  we  find  two  independent  integrals  u  ^=  a, 
V  =  b  o(  the  auxiliary  system  (3),  (sometimes  called  Lagrange's 
equations,)  and  then  put 

u  =  (p{v),  (4) 

an  equation  which  is  evidently  equally  general  with  /(^,  v)  =  o. 
Conversely,  it  may  be  shown  that  any  equation  of  the  form 
(4),  regarded  as  a  primitive,  gives  rise  to  a  definite  partial 
differential  equation  of  Lagrange's  linear  form.  For,  taking 
partial  derivatives  with  respect  to  the  independent  variables 
X  and  /,  we  have 


du       du 

a^  +  a^^ 

du       du 
dy  "^a^^ 


=  0X^O[; 


a'^+ai^J^ 


.,  Sdv    ,  dv 


■dy  '  ds' 


\. 


and  eliminating  (t)'{v)  from  these  equations,  the  term  contain- 
ing/^ vanishes,  giving  the  result 


du  du 
dy  ds 

dv  dv 
^d^ 

P  + 

du  du 
dz  dx 

dv  dv 
dz  dx 

^  = 

dudu 
dxdy 

dv  dv 

dxdy 

(5) 


which  is  of  the  form  Pp  -\-  Qq  —  R.\ 


*  Each  line  of  the  system  is  characterized  by  special  values  of  a  and  b  which 
we  may  call  its  coordinates,  and  the  surface  passes  through  those  lines  whose 
coordinates  are  connected  by  the  perfectly  arbitrary  relation /(a,  b)  =  C. 

f  These  values  of  P,  Q  and  R  are  known  as  the  "  Jacobians  "  of  the  pair 
of  functions  u,  v  with  respect  to  the  pairs  of  variables  y,  z\  z,  x ',  and  x,  y  re- 
spectively. Owing  to  their  analogy  to  the  derivatives  of  a  single  function  they 
are  sometimes  denoted  thus  : 


P  = 


d{ 


u,  v) 


d(u,  v) 


p  = 


d(u,  V) 


a(  r,  3)         a(3,  x)'    '    d{x,  y^ 

The  Jacobian  vanishes  if  the  functions  u  and  v  are  not  independent,  that  is 
to  say,  if  u  can  be  expressed  identically  as  a  function  of  v.     In  like  manner, 


-66  DIFFERENTIAL   EQUATIONS. 

As  an  illustration,  let  the  given  partial  differential  equa- 
tion be 

{mjs  —  ny)p  +  («^  —  lz)Q  =  ^y  ~  ^^t  (6) 

for  which  Lagrange's  Equations  are 

dx       _       dy       _       dz 
mz  —  ny~~  nx  —  Iz  ~  ly  —  mx'  ^'^ 

These  equations  were  solved  at  the  end  of  Art.  12,  the  two 
integrals  there  found  being 

Ix -\- my  ■\- nz  =  a     and     ;r'  +  y  +  ^^  =  b",  (8) 

Hence  in  this  case  the  system  of  **  Lagrangean  lines"  con- 
sists of  the  entire  system  of  circles  having  the  straight  line 

for  axis.     The  general  integral  of  equation  (6)  is  then 

lx-\-my  +  nz  =  <p{x'  +  /  +  ^'),  ( lo) 

which  represents  any  surface  passing  through  the  circles  just 
mentioned,  that  is,  any  surface  of  revolution  of  which  (9)  is  the 
axis.* 

Lagrange's  solution  extends  to  the  linear  equation  contain- 
ing n  independent  variables.     Thus  the  equation  being 

the  auxiliary  equations  are 

dx^      dx^  _         __  dx„      dz 

~-f-i — ! — -'  =  o  is  the  condition  that  <p  (a  function  of  x,  y  and  2)  is  expressible 

0{x,  y,  2) 

identically  as  a  function  of  u  and  v,  that  is  to  say,  that  0  =  o  shall  be  an  in- 
tegral of  Pp-\-  Qg  =  R. 

*  When  the   equation  Pdx  -\-  Qdy  +  Rdz  =  o  is  integrable  (as  it  is  in  the 
above  example;  see  Prob.  76,  Art.  20),  its  integral,  which  may  be  put  in  the  form 

V  =■  C,  represents  a  singly  infinite  system  of  surfaces  which  the  Lagrangean 
lines  cut  orthogonally  ;  therefore,  in  this  case,  the  general  integral  may  be  de- 
fined as  the  general  equation  of  the  surfaces  which  cut  orthogonally  the  system 

F  =  C.  Conversely,  starting  with  a  given  system  V  =  C,  u  ^=-  f{v)  is  the  gen- 
:eral  equation  of  the  orthogonal  surfaces,  \i  u  =  a  and  z/  =  3  are  integrals  of 


/  ^x        ^  I   "dy  I  dz 


COMPLETE    AND    GENERAL    INTEGRALS.  67 

and  if  u^  —  c^,  u^=z  c^,  ,  .  .  u^  =  c^  are  independent   integrals, 
the  most  general  solution  is 

fill,,  u^,  ,  .  .  Un)  =  o, 
where /is  an  arbitrary  function. 

Prob.  80.   Solve  xz- — h/^-^  =  ^«       Ans.  xy—  ^  =/(—]. 

Prob.  81.  Solve  i^y  +  z)p  +  {z -\-  x)q  —  x-\r  y. 
Prob.  82.  Solve  {x  -{- y){p  —  q)  —  z. 

Ans.  {x  +J^')  log  z  —  x  =  f{x  -\-y). 
Prob.  83,  Solve  x{y  —  z)p  -{-y{z  —  x)q  =  z{x  —y). 

Ans.  X  -\-y  -^  z=  f{xyz). 

Art.  22.    Complete  and  General  Integrals. 

We  have  seen  in  the  preceding  article  that  an  equation  be- 
tween three  variables  containing  an  arbitrary  function  gives 
rise  to  a  partial  differential  equation  of  the  linear  form.  It 
follows  that,  when  the  equation  is  not  linear  in  /  and  q,  the 
general  integral  cannot  be  expressed  by  a  single  equation  of 
the  form  cf){u,  v)  =  o;  it  will,  however,  still  be  found  to  depend 
upon  a  single  arbitrary  function. 

It  therefore  becomes  necessary  to  consider  an  integral  hav- 
ing as  much  generality  as  can  be  given  by  the  presence  of  arbi- 
trary constants.  Such  an  equation  is  called  a  "  complete  in- 
tegral "  ;  it  contains  two  arbitrary  constants  {n  arbitrary  con- 
stants in  the  general  case  of  n  independent  variables),  because 
this  is  the  number  which  can  be  eliminated  from  such  an  equa- 
tion, considered  as  a  primitive,  and  its  two  derived  equations. 
For  example,  if 

{x-ay  +  {y-br  +  z'=^k\ 

a  and  b  being  regarded  as  arbitrary,  be  taken  as  the  primitive, 
the  derived  equations  are 

X  —  a  -\-  zp  =.  O,        y  —  b  -\-  zg  =  0, 
and  the  elimination  of  a  and  b  gives  the  differential  equation 

of  which  therefore  the  given  equation  is  a  complete  integral. 


58  DIFFERENTIAL   EQUATIONS. 

Geometrically,  the  complete  integral  represents  a  doubly  in- 
finite system  of  surfaces  ;  in  this  case  they  are  spherical  sur- 
faces having  a  given  radius  and  centers  in  the  plane  of  xy. 

In  general,  a  partial  differential  equation  of  the  first  order 
with  two  independent  variables  is  of  the  form 

F{x,y,z,p,q)=0,  (I) 

and  a  complete  integral  is  of  the  form 

A^^  yy  ^'  ^'  ^)  =  o-  (2) 

In  equation  (i)  suppose  x,  y  and  z  to  have  special  values, 
namely,  the  coordinates  of  a  special  point  A  ;  the  equation 
becomes  a  relation  between  p  and  q.  Now  consider  any  sur- 
face passing  through  A  of  which  the  equation  is  an  integral  of 
(i),  or,  as  we  may  call  it,  a  given  ''integral  surface  "  passing 
through  A.  The  tangent  plane  to  this  surface  at  A  determines 
values  of  p  and  q  which  must  satisfy  the  relation  just  men- 
tioned. Consider  also  those  of  the  complete  integral  surfaces 
[equation  (2)]  which  pass  through  A.  They  form  a  singly  in- 
finite system  whose  tangent  planes  at  A  have  values  of  p  and 
q  which  also  satisfy  the  relation.  There  is  obviously  among 
them  one  which  has  the  same  value  of  p,  and  therefore  also 
the  same  value  of  q,  as  the  given  integral.  Thus  there  is  one 
of  the  complete  integral  surfaces  which  touches  at  A  the  given 
integral  surface.  It  follows  that  every  integral  surface  (not  in- 
cluded in  the  complete  integral)  must  at  every  one  of  its  points 
touch  a  surface  included  in  the  complete  integral.* 

It  is  hence  evident  that  every  integral  surface  is  the  en- 
velope of  a  singly  infinite  system  selected  from  the  complete 
integral  system.  Thus,  in  the  example  at  the  beginning  of 
this  article,  a  right  cylinder  whose  radius  is  k  and  whose  axis 
lies  in  the  plane  of  xy  is  an  integral,  because  it  is  the  envelope 

*  Values  of  x,y,  and  z,  determining  a  point,  together  with  values  of/  and  f, 
determining  the  direction  of  a  surface  at  that  point,  are  said  to  constitute  an 
"element  of  surface."  The  theorem  shows  that  the  completfe  integral  is  'com- 
plete "  in  the  sense  of  including  all  th6  surface  elements  which  satisfy  the  differ- 
ential equation.  The  method  of  grouping  the  "consecutive"  elements  to  form 
an  integral  surface  is  to  a  certain  extent  arbitrary. 


COMPLETE    AND    GENERAL    INTEGRALS.  59^ 

of  those  among  the  spheres  represented  by  the  complete  in- 
tegral whose  centers  are  on  the  axis  of  the  cylinder.  If  we 
make  the  center  of  the  sphere  describe  an  arbitrary  curve  in 
the  plane  of  xy  we  shall  have  the  general  integral  in  this  ex- 
ample. 

In  general,  if  in  equation  (2)  an  arbitrary  relation  between^ 
a  and  b,  such  as  <^  =  0(^)j  be  established,  the  envelope  of  the 
singly  infinite  system  of  surfaces  thus  defined  will  represent 
the  general  integral.  By  the  usual  process,  the  equation  of 
the  envelope  is  the  result  of  eliminating  a  between  the  two 
equations 

f{x,  y,  z,  a,  (p{a)  )  =  o,     ^/(-^^  J,  ^,  «,  0(«)  )  =  O.         (3) 

These  two  equations  together  determine  a  line,  namely,  the 
"  ultimate  intersection  of  two  consecutive  surfaces."  Such 
lines  are  called  the  "  characteristics  "  of  the  differential  equa- 
tion. They  are  independent  of  any  particular  form  of  the 
complete  integral,  being  in  fact  lines  along  which  all  integral 
surfaces  which  pass  through  them  touch  one  another.  In  the 
illustrative  example  above  they  are  equal  circles  with  centers 
in  the  plane  of  xy  and  planes  perpendicular  to  it.* 

The  example  also  furnishes  an  instance  of  a  "  singular  so- 
lution "  analogous  to  those  of  ordinary  differential  equations. 

*The  characteristics  are  to  be  regarded  not  merely  as  lines,  but  as  **  linear 
elements  of  surface,"  since  they  determine  at  each  of  their  points  the  direction 
of  the  surfaces  passing  through  them.  Thus,  in  the  illustration,  they  are  cir- 
cles regarded  as  great-circle  elements  of  a  sphere,  or  as  elements  of  a  right 
cylinder,  and  may  be  likened  to  narrow  hoops.  They  constitute  in  all  cases  a 
triply  infinite  system.  The  surfaces  of  a  complete  integral  system  contain  them 
all,  but  they  are  differently  grouped  in  different  integral  surfaces. 

If  we  arbitrarily  select  a  curve  in  space  there  will  in  general  be  at  each  of 
its  points  but  one  characteristic  through  which  the  selected  curve  passes;  that 
is,  whose  tangent  plane  contains  the  tangent  to  the  selected  curve.  These  char- 
acteristics (for  all  points  of  the  curve)  form  an  integral  surface  passing  through 
the  selected  curve  ;  and  it  is  the  only  one  which  passes  through  it  unless  it  be 
itself  a  characteristic.  Integral  surfaces  of  a  special  kind  result  when  the  se- 
lected curve  is  reduced  to  a  point.  In  the  illustration  these  are  the  results  of 
rotating  the  circle  about  a  line  parallel  to  the  axis  of  z. 


60  DIFFERENTIAL    EQUATIONS. 

For  the  planes  z  =  ±  k  envelop  the  whole  system  of  spheres 
represented  by  the  complete  integral,  and  indeed  all  the  sur- 
faces included  in  the  general  integral.  When  a  singular  solu- 
tion exists  it  is  included  in  the  result  of  eliminating  a  and  b 
from  equation  (2)  and  its  derivatives  with  respect  to  a  and  b^ 
that  is,  from 

but,  as  in  the  case  of  ordinary  equations,  this  result  may  in- 
clude relations  which  are  not  solutions. 

Prob.  84.  Derive  a  differential  equation  from  the  primitive 
Ix  +  ffiy  -\-  nz  =^  a,  where  /,  m,  n  are  connected  by  the  relation 
P-\-m^-\-fi'=  I. 

Prob.  85.  Show  that  the  singular  solution  of  the  equation 
found  in  Prob.  84  represents  a  sphere,  that  the  characteristics  con- 
sist of  all  the  straight  lines  which  touch  this  sphere,  and  that  the 
general  integral  therefore  represents  all  developable  surfaces  which 
touch  the  sphere. 

Prob.  86.  Find  the  integral  which  results  from  taking  in  the 
general  integral  above  /'  +/«''  =  cos'  0  (a  constant)  for  the  arbitrary 
relation  between  the  parameters. 

Art,  23.     Complete  Integral  for  Special  Forms. 

A  complete  integral  of  the  partial  differential  equation 

F{x,y,  z,p,  q)  =  0  (i) 

contains  two  constants,  a  and  b.  If  a  be  regarded  as  fixed  and 
b  as  an  arbitrary  parameter,  it  is  the  equation  of  a  singly  in- 
finite system  of  surfaces,  of  which  one  can  be  found  passing 
through  any  given  point.  The  ordinary  differential  equation 
of  this  system,  which  will  be  independent  of  b,  may  be  put  in 

the  form 

dz  =  pdx  +  qdy,  (2) 

in  which  the  coefficients/  and  q  are  functions  of  the  variables 
and  the  constant  a.  Now  the  form  of  equation  (2)  shows  that 
these  quantities  are  the  partial  derivatives  of  ^,  in  an  integral 
of  equation  (i);  therefore  they  are  values  of  /  and  q  which 


COMPLETE    INTEGRAL    FOR    SPECIAL    FORMS.  61 

satisfy  equation  (i).  Conversely,  if  values  of/  and  q  in  terms 
of  the  variables  and  a  constant  a  which  satisfy  equation  (i)  are 
such  as  to  make  equation  (2)  the  differential  equation  of  a  sys- 
tern  of  surfaces,  these  surfaces  will  be  integrals.  In  other 
words,  if  we  can  find  values  of/  and  q  containing  a  constant  a 
which  satisfy  equation  (i)  and  make  dz  =  pdx -\- qdy  inte^ 
grable,  we  can  obtain  by  direct  integration  a  complete  inte- 
gral, the  integration  introducing  a  second  constant. 

There  are  certain  forms  of  equations  for  which  such  values 
of  p  and  q  are  easily  found.  In  particular  there  are  forms  in 
which  p  and  q  admit  of  constant  values,  and  these  obviously 
make  equation  (2)  integrable.  Thus,  if  the  equation  contains 
/  and  q  only,  being  of  the  form 

^(A^)  =  o,  (3) 

we  may  put/  =  a  and  q  =  b,  provided 

F{a,b)  =  o.  (4) 

Equation  (2)  thus  becomes 

dz  =  adx  -\-  bdy, 
whence  we  have  the  complete  integral 

z  =^  ax  -{-  by  -{-  c,  (5) 

in  which  a  and  b  are  connected  by  the  relation  (4)  so  that  a,  b 
and  c  are  equivalent  to  two  arbitrary  constants. 

In  the  next  place,  if  the  equation  is  of  the  form 

2=px  +  qy+f{p,q),  .       (6) 

which  is  analogous  to  Clairaut's  form,  Art.  10,  constant  values' 
of/  and  q  are  again  admissible  if  they  satisfy 

3  =  ax  +  by-^f{a,b\  (;) 

and  this  is  itself  the  complete  integral.  For  this  equation  is 
of  the  form  2  =  ax  -{-  by  -\-  c,  and  expresses  in  itself  the  rela- 
tions between  the  three  constants.  Problem  84  of  the  preced- 
ing article  is  an  example  of  this  form. 

In  the  third  place,  suppose  the  equation  to  be  of  the  form 
F{z,p,q)  =  o,  (8) 


€2  DIFFERENTIAL    EQUATIONS. 

in  which  neither  ;rnor_;/  appears  explicitly.  If  we  assume 
q  =  apy  p  will  be  a  function  of  z  determined  from 

F{z,  /,  ap)  =  o,     say    p  =  (piz).  (9) 

Then  dz  —pdx  -f-  qdy  =  o  becomes  dz  =  <p{z)(dx  -j-  adjf)f  which  is 
integrable,  giving  the  complete  integral 

A  fourth  case  is  that  in  which,  while  z  does  not  explicitly 
occur,  it  is  possible  to  separate  x  and  /  from  /  and  ^,  thus  put- 
ting the  equation  in  the  form 

fk^.P)=ny^q)'  (li) 

If  we  assume  each  member  of  this  equation  equal  to  a  con- 
stant ^,  we  may  determine/  and  q  in  the  forms 

/  =  <t>kx,  a\  q  =  (Ply,  a).  (12) 

and  dz  =  pdx  +  qdy  takes  an  integrable  form  giving 

-  =  y  VX-^.  ^y^  +  J<P.{y^  OL)dy  +  ^.  (13) 

It  is  frequently  possible  to  reduce  a  given  equation  by  trans- 
formation of  the  variables  to  one  of  the  four  forms  considered 
in  this  article.*  For  example,  the  equation  x'^p'  -\- y"^  q^  z=  z" 
xnay  be  written 

*The  general  method,  due  to  Charpit,  of  finding  a  proper  value  oip  consists 
of  establishing,  by  means  of  the  condition  of  integrability,  a  linear  partial  dif- 
ferential equation  for/,  of  which  we  need  only  a  particular  integral.  This  may 
be  any  value  of  p  taken  from  the  auxiliary  equations  employed  in  Lagrange's 
process.  See  Boole,  Differential  Equations  (London  1865),  p.  336  ;  also  For- 
syth, Differential  Equations  (London  1885),  p.  316,  in  which  the  auxiliary  equa- 
tions are  deduced  in  a  more  general  and  symmetrical  form,  involving  both  / 

.  and  q.  These  equations  are  in  fact  the  equations  of  the  characteristics  regarded 
as  in  the  concluding  note  to  the  preceding  article.     Denoting  the  partial  deriva- 

riives  of  F{x,  y,  z,  p,  q)  by  X,  V,  Z.  P,  Q,  they  are 

dx   _   dv   _         d^        _  ^^       _  ^9 

'P'  -~Q~   Pp^Qq  ~  ~  'X+Z^  -~   Y^Zq 

See  Jordan's  Cours  d'Analyse  (Paris,  1887),  vol.  in,  p.  318  ;  Johnson's  Differ- 
ential Equations  (New  York,  i88g),  p.  300.  Any  relation  involving  one  or  both 
the  quantities  /  and  q,  combined   with  F^o,  will  furnish  proper  values  of  / 


PARTIAL    EQUATIONS,    SECOND    ORDER.  63 

whence,  putting  x'  =  log  x,  y  =  logj,  ^'  =  log  z,  it  becomes 
/  '  +^'"  =  ij  which  is  of  the  form  F{p\  q')  =  o,  equation  (3). 
Hence  the  integral  is  given  by  equation  (5)  when  a''  -^  d^  =  i; 
it  may  therefore  be  written 

y  =  x'  cos  a  -\-  y'  sin  «  +  <:, 
and  restoring  X^  y^  and  z,  that  of  the  given  equation  is 

Prob.  87.  Find  a  complete  integral  for/'  —  q^  =  i, 

Ans.  z  =z  X  sec  a  -^  y  tan  a  +  <5. 
Prob.  88.  Find  the  singular  solution  of  z  ^  px  +  qy  +/^' 

Ans.  2  =  —  jTv. 
Prob.  89.  Solve  by  transformation  q  —  2yp^. 

Ans.  z  =  ax  -\~  a^y^  +  b. 
Prob.  90.  Solve  z{p'—q')  =  x  —  y. 

Ans.  2^  =(x  +  a)^  +(7  +  ^)^  +  ^.   " 
Prob.  91.   Show  that  the  solution  given  for  the  form  J^(z,p,  q)  —  o 
represents   cylindrical  surfaces,  and  that  J*'(z,  o,  o)  =  o  is  a  singular 
solution. 

Prob.  92.  Deduce  by  the  method  quoted  in  the  foot-note  two 
complete  integrals  of  pq  =  px  -{-  qy. 

Ans.   2z=l-^-{-  ocyj  4-y5,  and  z  =  xy -\- y  \/{x''  -^  a)  +  b. 

Art.  24.    Partial  Equations  of  Second  Order. 

We  have  seen  in  the  preceding  articles  that  the  general 
solution  of  a  partial  differential  equation  of  the  first  order  de- 
pends upon  an  arbitrary  function  ;  although  it  is  only  when 
the  equation  is  linear  in  /  and  q  that  it  is  expressible  by  a 
single  equation.  But  in  the  case  of  higher  orders  no  general 
account  can  be  given  of  the  nature  of  a  solution.  Moreover, 
when  we  consider  the  equations  derivable  from  a  primitive  con- 
taining arbitrary  functions,  there  is  no  correspondence  between 
their  number  and  the  order  of  the  equation.     For  example,  if 

and  q.     Sometimes  several  such  relations  are  readily  found  ;  for  example,  for 
the  equation  z  =  pq  we  thus  obtain  the  two  complete  integrals 

z  =  {y-\-a){x-^b)     and     42  =  ^£  +  ^ry  +  ysV . 


04  DIFFERENTIAL    EQUATIONS. 

the  primitive  with  two  independent  variables  contains  two  ar- 
bitrary functions,  it  is  not  generally  possible  to  eliminate  them 
and  their  derivatives  from  the  primitive  and  its  two  derived 
equations  of  the  first  and  three  of  the  second  order. 

Instead  of  a  primitive  containing  two  arbitrary  functions^ 
let  us  take  an  equation  of  the  first  order  containing  a  single 
arbitrary  function.  This  may  be  put  in  the  form  u  =  (p{v), 
u  and  V  now  denoting  known  functions  of  x,  jy,  je;, />,  a.nd  q. 
<p'{y)  may  now  be  eliminated  from  the  two  derived  equations 
as  in  Art.  21.     Denoting  the  second  derivatives  of  z  by 

the  result  is  found  to  be  of  the  form 

Rr-\-Ss-\-Tt-^U{rt^  $")=¥,  (i) 

in  which  R^  S,  T,  [/,  and  V  are  functions  of  x,  y,  ^,  /,  and  q. 
With  reference  to  the  differential  equation  of  the  second  order 
the  equation  u  =  0(z/)  is  called  an  "  intermediate  equation  of 
the  first  order":  it  is  analogous  to  the  first  integral  of  an  ordi- 
nary equation  of  the  second  order.  It  follows  that  an  inter- 
mediate equation  cannot  exist  unless  the  equation  is  of  the 
form  (i);  moreover,  there  are  two  other  conditions  which 
must  exist  between  the  functions  R,  5,  Z",  and  U, 

In  some  simple  cases  an  intermediate  equation  can  be  ob- 
tained by  direct  integration.  Thus,  if  the  equation  contains 
derivatives  with  respect  to  one  only  of  the  variables,  it  may  be 
treated  as  an  ordinary  differential  equation  of  the  second  order, 
the  constants  being  replaced  by  arbitrary  functions  of  the 
other  variable.  Given,  for  example,  the  equation  xr  —  p  ^=  xy, 
which  may  be  written 

'  xdp  —  pdx  —  xy  dx. 

This  becomes  exact  with  reference  to  x  when  divided  by  x^^ 
and  gives  the  intermediate  equation 

p=yx\ogx-\rX(}){y\ 
A  second  integration  (and  change  in  the  form  of  the  arbitrary 
function)  gives  the  general  integral 

z  =  iyx'  log  X  -h  x'(p{y)  +  tp{y). 


PARTIAL   EQUATIONS,    SECOND    ORDER,  65 

Again,  the  equation  p-\-  r  +s  =  i  is  already  exact,  and 
gives  the  intermediate  equation 

which  is  of  Lagrange's  form.     The  auxihary  equations*  are 

^  ^^ 

of  which  the  first  gives  x  —  j^  =^  a,  and  eHminating  x  from  the 
second,  its  integral  is  of  the  form 

Hence,  putting  ^  =  tp{a),  we  have  for  the  final  integral 

2  =  x  +  0(j/)  +  e  -yxp(x  -  y\ 
in  which  a  further  change  is  made  in  the  form  of  the  arbitrary 
function  0. 

Prob.  93.  Solve  t  —  q  =^  e^  -\-  e^. 

Ans.  z  =y(ey  -  ^^)  +  (p{x)  +  eyip(x). 

Prob.  94.  Solve  r  -{-/  =y\ 

Ans.  z  =  logle'^ytpiy)  -  ^" "'^l  +  Hy)- 

Prob.  95.  Solve  y(j  —  t)  z=:  x. 

Ans.  z={x-\-y)  \ogy  +  (p(x)  +  ^(^  -^  y). 

Prob.  96.  Solve  ps  —  qr  =  o.  Ans.  x  =  (p(y)  +  ^(2). 

Prob.  97.  Show  that  Monge's  equations  (see  foot-note)  give  for 
Prob.  96  the  intermediate  integral  /  =  (p{z)  and  hence  derive  the 
solution. 

*In  Monge's  method  (for  which  the  reader  must  be  referred  to  the  complete 
treatises)  of  finding  an  intermediate  integral  of 

i?r  +  5j  +  TV  =  F 
when  one  exists,  the  auxiliary  equations 

J^df  -  Sdy  dx  4-  Tdx'^  =  o,         Rdp  dy  +  Tdq  dx  =  Vdx  dy 
are  established.     These,  in  connection  with 

dz  =  pdx  -j-  qdy, 
form  an  incomplete  system  of  ordinary  differential  equations,  between  the  five 
variables  x,  y,  z,  p,  and  q.  But  if  it  is  possible  to  obtain  two  integrals  of  the 
system  in  the  form  u  =:  a,  v  =  b,  u  =  (p{v)  will  be  the  intermediate  integral. 
The  first  of  the  auxiliary  equations  is  a  quadratic  giving  two  values  for  the  ratio 
dy.dx.  If  these  are  distinct,  and  an  intermediate,  integral  can  be  found,  for 
each,  the  values  of  p  and  q  determined  from  them  will  make  dz  =pdx-\-qdy 
jntegrab!e,  and  give  the  general  integral  at  once. 


()()  DIFFERENTIAL   EQUATIONS. 

Prob.  98.  Derive  by  Monge's  method  for  q^r  —  2pqs  +/'/  =  o 
the  intermediate  integral/  =  ^0(2),  and  thence  the  general  integral. 

Ans.  y  +  x(t>{z)  =  ^-(s). 

Art.  25.    Linear  Partial  Differential  Equations. 

Equations  which  are  linear  with  respect  to  the  dependent 
variable  and  its  partial  derivatives  may  be  treated  by  a  method 
analogous  to  that  employed  in  the  case  of  ordinary  differential 
equations.  We  shall  consider  only  the  case  of  two  independ- 
ent variables  x  and  j/,  and  put 

so  that  the  higher  derivatives  are  denoted  by  the  symbols  Z?', 
DD\  D"*y  D^y  etc.  Supposing  further  that  the  coefBcients  are 
constants,  the  equation  may  be  written  in  the  form 

f{D,D')z  =  F{x,y),  (I) 

in  which  /  denotes  an  algebraic  function,  or  polynomial,  of 

which  the  degree  corresponds  to  the  order  of  the  differential 

equation.     Understanding  by  an  "integral"  of  this  equation 

an  explicit  value  of  z  in  terms  of  x  and  y,  it  is  obvious,  as  in 

Art.  15,  that  the  sum  of  a  particular  integral  and  the  general 

integral  of 

AD,  D')z  =  o  (2) 

will  constitute  an  equally  general  solution  of  equation  (i).  It 
is,  however,  only  when/(Z>,  D')  is  a  homogeneous  function  of  D 
and  D'  that  we  can  obtain  a  solution  of  equation  (2)  containing 
71  arbitrary  functions,*  which  solution  is  also  the  "comple- 
mentary function"  for  equation  (i). 

Suppose  then  the  equation  to  be  of  the  form 

and  let  us  assume  z  =  (fj{y  +  ^^)y  (4) 

*  It  is  assumed  that  such  a  solution  constitutes  the  general  integral  of  an 
equation  of  the  wth  order;  for  a  primitive  containing  more  than  n  independent 
arbitrary  functions  cannot  give  rise  by  their  elimination  to  an  equation  of  the 
wth  order. 


LINEAR    PARTIAL    EQUATIONS.  67 

where  m  is  a.  constant  to  be  determined.  From  equation  (4), 
D^  =  m(p'{y  +  w;r)  and  D'z  =  (p\y  -j-  mx)^  so  that  Dz  =  mD'Zy 
Vz  =  m^U'^z,  DD'z  =  mD'^'y,  etc.  Substituting  in  equation  (3) 
and  rejecting  the  factor  D'^'z  or  (fp'\y  +  ^^-^)»  we  have 

A^m''  +  A.nf"-'  +  .  .  .  +  ^„  =  o  {5) 

for  the  determination  of  m.  If  m^, ;//,,  .  .  .  m^  are  distinct  roots 
of  this  equation, 

-       Z  =  (pXy  +  ^^1^)  +0a(7  +  ^^2-^)  +  •  •   •  +  0«(  J  +  ^n^)     (6) 

is  the  general  integral  of  equation  (3). 

For  example,  the  general  integral  of  — ,  —  --.,  =  o  is  thus 

found  to  be  ^  =  cf){y  +  ^)  +  ^p{y  —  •^).  Any  expression  of  the 
form  Axy  -\-  Bx  -\-  Cy  -\-  D  '\s  2.  particular  integral ;  accordingly 
it  is  expressible  as  the  sum  of  certain  functions  of  x-\-y3ind 
X  —  y  respectively. 

The  homogeneous  equation  (3)  may  now  be  written  sym- 
bolically in  the  form 

{D  -  m,D'){D  -  m,D') .  .  .  {D  -  m„D')z  =  o,  (7) 

in  which  the  several  factors  correspond  to  the  several  terms  of 
the  general  integral.  If  two  of  the  roots  of  equation  (5)  are 
■equal,  say,  to  ;;/,,  the  corresponding  terms  in  equation  (6)  are 
equivalent  to  a  single  arbitrary  function.  To  form  the  general 
integral  we  need  an  integral  of 

{D  -  m^DJz  :=  o  (8) 

in  addition  to  0(j/  +  m^x).     This  will  in  fact  be  the  solution  of 

(D  —  mfi')z  =  (p{y  -f  m^x);  (9) 

for,  if  we  operate  with  D  —  m^D'  upon   both  members  of  this 

equation,  we  obtain  equation  (8).     Writing  equation  (9)  in  the 

form 

p  —  m^q  =  (p{y  -\-  mx)y 

Lagrange's  equations  are 

dy  _  dz 


giving  the  integrals  y  -\-  m^x  =^  a,  z  :=^  x(p(a)  -f-  b.     Hence  the 
integral  of  equation  (9)  is 

2  =  x(p(y  +  m^x)  +  ip{y  -{-  m^x),  (10) 


68  DIFFERENTIAL   EQUATIONS. 

and  regarding  0  also  as  arbitrary,  these  are  the  two  independ- 
ent terms  corresponding  to  the  pair  of  equal  roots. 

If  equation  (5)  has  a  pair  of  imaginary  roots  rn  =■  fi  ±,  iVy 
the  corresponding  terms  of  the  integral  take  the  form 

(t){f-]-^x  -\-ivx)  +  tp{y  +  M^  —  iyx)y 

which  when  0  and  tp  are  real  functions  contain  imaginary 
terms.  If  we  restrict  ourselves  to  real  integrals  we  cannot 
now  say  that  there  are  two  radically  distinct  classes  of  inte- 
grals ;  but  if  any  real  function  oi  y  -{-  fix  ~\-  ivx  be  put  in  the- 
form  X-\-iY,  either  of  the  real  functions  X  or  F  will  be  an 
integral  of  the  equation.     Given,  for  example,  the  equation 

of  which  the  general  integral  is^ 

2  =  (li{y  +  ix)  +  ^{y  —  ix) ; 
to  obtain  a  real  integral  take  either  the  real  or  the  coefficient 
of  the  imaginary  part  of  any  real  form  of  0(j^  +  ix).     Thus,  if 
0(/)  =  ^  we  find  e^  qosx  and  e^  sin;jr,  each  of    which    is   an 
integral  (see  Chap.  VI,  p.  245). 

As  in  the  corresponding  case  of  ordinary  equations,  the 
particular  integral  of  equation  (i)  may  be  made  to  depend 
upon  the  solution  of  linear  equations  of  the  first  order.     The 

inverse  symbol  jz jj,^(x,  y)  in  the  equation  corresponding^ 

to  equation  (14),  Art.  16,  denotes  the  value  of  z  in 

{D  —  fnD')2  =  F{x,  y)     or    p  —  mq=F{x,y),  (li) 

For  this  equation  Lagrange's  auxiliary  equations  give 

y  -\-  mx  =  ^,      ^  ^  I  F(p^^  ^  ~  ff^x)^-^  +  ^  =  ■^i(^»  ^)  +  ^» 
and  the  general  integral  is 

2  =  F^{x,y  +  mx)  +  (p(y  +  mx),  (12) 

The  first  term,  which  is  the  particular  integral,  may  there- 
fore be  found  by  subtracting  7nx  from  y  in   F{x,  y),  inte- 


LINEAR    PARTIAL   EQUATIONS.  69 

grating  with  respect  to  x,  and  then  adding  mx  to  y  in  the 
result.* 

For  certain  forms  of  F{x,  y)  there  exist  more  expeditious 
methods,  of  which  we  shall  here  only  notice  that  which  applies 
to  the  form  F{ax  +  by).  Since  DF{ax  -j-  by)  =  aF\ax  +  by) 
and  D'F{ax -\- by)  =:  bF\ax -\-by),  it  is  readily  inferred  that, 
when  fip,  D')  is  a  homogeneous  function  of  the  «th  degree, 

/(A  D')F{ax  +  by)  =f{a,  b)F^-\ax  +  by).  (13) 

That  is,  if  t  ^^  ax  -\-  by,  the  operation  of  /(A  D')  on  F{t)  is 
equivalent  to  multiplication  hy  f{a,b)  and  taking  the  nth.  de- 
rivative, the  final  result  being  still  a  function  of  t.  It  follows 
that,  conversely,  the  operation  of  the  inverse  symbol  upon  a 
function  of  /  is  equivalent  to  dividing  by /(^,  b)  and  integrating 
n  times.     Thus, 

^^Pia.  +  by)=^ff...fF(t)dr.       (.4) 

When  ax  -\-by  \s  di  multiple  oi  y  -{-  m^x,  where  m^  is  a  root  of 
equation  (5),  this  method  fails  with  respect  to  the  correspond- 
ing symbolic  factor,  giving  rise  to  an  equation  of  the  form  (9), 
of  which  the  solution  is  gi\i|en  in  equation  (10).  Given,  for  ex- 
ample, the  equation 

^  +  SJd^  -  "  d/  =  ^^"  (^  - -^)  +  ^^"  (^  +-^> 
or         {D  —  U)  {D  +  2D')3  =  s\n(x  —y)-\-  sin  {x  -f  y). 
The  complementary  function  is  (p(y  -\-  x) -\-  tp(y  —  2x).     The 
part  of  the  particular  integral  arising  from  sin  {x  —  y),  in  which 

^  =  I,  ^  =  —  I,  is /    /sin  tdf  =  -  sin  {x  —  y).     That  aris- 

*  The  symbolic  form  of  this  theorem  is 

D-mD'^^''^  y^  =  '"^^^'f'-  '«^^'^(^,  y)dx 
corresponding  to  equation  (13),  Art.  16.     The  symbol  e^xD'  here  indicates  the 
addition  of  mx  to  y  in  the  operand.     Accordingly,  using  the  expanded  form 
of  the  symbol, 

ern^D'F^y)  =  (I  +  ;;,^  i.  +  ^'  |!  4-  . .  .)  F(,y)  =  F{y  +  fnx\ 

the  symbolic  expression  of  Taylor's  Theorem. 


70  DIFFERENTIAL   EQUATIONS. 

ing  from  sin  {x  -f-^)  which  is  of  the  form  of  a  term  in  the  com- 
plementary function  is jz jr,  cos  (x  -f^'),  which  by  equa- 
tion (lo)  is  —  J^^r  cos  (x-{-y).     Hence  the  general  integral  of 
the  given  equation  is 
2  =  0(7  +  x)  +  tp{y  -  2x)+\  sin  {x  -  y)  -  :^  x  cos  {x  +7).. 

If  in  the  equation /(Z?,  D')z  z^  o  the  symbol /(Z>,  D'\  though 
not  homogeneous  with  respect  to  D  and  D\  can  be  separated 
into  factors,  the  integral  is  still  the  sum  of  those  corresponding 
to  the  several  symbolic  factors.  The  integral  of  a  factor  of 
the  first  degree  is  found  by  Lagrange's  process ;  thus  that  of 

{D  —  mD'  —  a)z  =  o  (15) 

is  2  =  e^(l){y  +  mx).  (16) 

But  in  the  general  case  it  is  not  possible  to  express  the 
solution  in  a  form  involving  arbitrary  functions.  Let  us,  how- 
ever, assume 

^  =  ^^^+*>',  (17) 

where  Cj  h,  and  k  are  constants.  Since  D^*^^^  ■=  h^*^^^ 
and  Z)'^+^=/^/*+.*^  substitution  in  f\D,  D')z  =  o  gives 
cf{h,  i)^+^  =  o.  Hence  we  have  a  solution  of  the  form  (17) 
whenever  h  and  k  satisfy  the  relation 

Ah,k)  =  o,  (18) 

c  being  altogether  arbitrary.  It  is  obvious  that  we  may  also 
write  the  more  general  solution 

^=  JS^r^^+^W-^,  (19) 

where  k  =  F{h)  is  derived  from  equation  (18),  and  c  and  h  admit 
of  an  infinite  variety  of  arbitrary  values. 

Again,  since  the  difference  of  any  two  terms  of  the  form 
gkx-\-F(h)y  with  different  values  of  k  is  included  in  expression 
(19),  we  infer  that  the  derivative  of  this  expression  with  respect 
to  h  is  also  an  integral,  and  in  like  manner  the  second  and 
higher  derivatives  are  integrals. 
For  example,  if  the  equation  is 

(£2      (h_ 
dx'~d^-^' 


LINEAR    PARTIAL    EQUATIONS.  71 


for  which  equation   (i8)  is  ^'  —  ^  =  o,  we  have  classes  of  in- 
tegrals of  the  forms  ^ 
^x  +  A»^  ^     ^x  +  AV(;tr  _|_  2hy), 


In  particular,  putting  /?  =  o  we  obtain  the  algebraic  integrals 
c^x,  c,(x'-{-2y),  c,{x'  +  6xy),  etc. 

The  solution  of  a  linear  partial  differential  equation  with 
variable  coefficients  may  sometimes  be  effected  by  a  change  of 
the  independent  variables  as  illustrated  in  some  of  the  exam- 
ples below. 

Prob.  99.  Show  that  if  m^  is  a  triple  root  the  corresponding 
terms  of  the  integral  are  x'^(l){y  -f-  m^x)  -{-  xip(y-\-  m^x)-{-  x{y-\-^i^)* 

Prob.  100.  Solve  2—^-  —  3;^-^ 2^—^  =  o. 

dx^       "^dxdy        dy 

Prob.  Id.  Solve  ^-^  +  2^^,  +  8/  =  ^• 

Ans.  z  =  <p{x)  +  ^(:r  +  j>;)  +  xx(x  -{-y)  —y  log  a:. 
'    Prob.  102.  Solve  {jD'  +  sDD'  +  6Z>")s  =  (jj'  -  2^)"^ 

Ans.  z  =  (p{y  —  2x)  +  tp(y  —  ^x)  +  x  log  (^  —  2x). 

'd'^z         'd^z        dz 

Prob.  103.   Solve  ^-^  —  :r-^r-  -{-^r z=  o. 

"^  9^       dxdy      dy 

Prob.  104.  Show  that  for  an  equation  of  the  form  (15)  the  solu- 
tion given  by  equation  (19)  is  equivalent  to  equation  (16). 

Prob.  105.  Solve  — —1 z^T'^—^^Tl 1^   ^7  transposi« 

X  ox      X  dx      y  dy      y  dy 

tion  to  the  independent  variables  x^  and  y. 

Prob.  106.  Solve  .'g  +  ..y^^  +yg-  =  o. 


INDEX. 


Auxiliary  system  of  lines,  51,  55- 

Besselian  functions,  44  note. 
Bessel's  equation,  43. 
Boundary  (of  real  solutions),  17. 

Characteristics,  59. 
Clairaut's  equation,  22,  61. 
Complementary  function,  35,  48. 
Complete  integral,  2,  30,  57,  60. 
Condition  of  integrability,  50. 
Cuspnlocus,  17,  20. 

Decomposable  equations,  13. 
Differentiation,  solution  by,  20. 
Direct  integration,  2. 
Discriminant,  16,  18. 
Doubly  infinite  systems,  26,  31. 

Envelope,  15. 

Equation  of  energy,  29,  49. 

Equipotential  surfaces,  52  note. 

Exact  differentials,  2,  51. 

Exact  equations,  6,  27,  30,  36. 

Extension  of  the  linear  equation,  12. 

Finite  solutions,  45 

First  integral,  30,  31. 

First  order  and  degree,  1,  50. 

First  order  and  second  degree,  12. 

General  integial,  53^  57. 
Geometrical  applications,  23,  31. 
Geometrical  representation  of  a  differ 
ential  equation,  3,  13,  15,  26. 

Homogeneous  equations,  9. 
Homogeneous  linear  equations,  40 

Integral,  26,  34,  49,  66. 
Integral  equation,  26. 
Integral  surface,  58. 
Integrating  factors,  8,  11,  33. 

Jacobians,  55  note. 


Lagrange's  lines,  55. 

Lagrange's  solution,  53,  56. 

Linear  elements  of  surface,  59  note. 

Linear  equations,  10,  34,  36. 

Linear  partial  differential  equation,  66. 

Logarithmic  solutions,  44  note,  46  note. 


note. 


Monge's  method,  65,  Prob.  97 
Monge's  solution,  53,  Prob.  79 

Node-locus,  19. 
Non-integrable  equation,  51. 

Operative  symbols,  36,  41. 
Order,  equations  of  first,  i. 

of  second,  28. 
Orthogonal  surfaces,  52  note. 
Orthogonal  trajectories,  24, 


Parameters  or  arbitrary  constants,  4, 

15,  26,  31,  60. 
Partial  differential  equations. 

first  order  and  degree,  53. 

linear,  66. 

second  order,  63 
Particular  integral.  2 

determined  in  series,  46. 

of  linear  equation,  35,  38.  41. 
Pencil  ot  curves,  14. 
Primitive  5.  55 

Separation  of  variables,  2,  51. 
Series,  solutions  in,  42. 
Simultaneous  equations,  25,  47 
Singular  solutions,  15,  18,  26  note,  59. 
Symbolic  solutions,  37  et  seq..  41,  67. 
Systems  of  curves,-^4,  26,  31. 
Systems  oi  differential  equations,  47. 

Tac  locus,  16 

Trajectories,  23 

Transcendental  and  algebraic  forms  of 

solution,  2. 
Transformation  of  hnear  equations,  46. 

Ultimate  intersections,  19,  59. 


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Barr's  Kinematics  of  Machinery Svo,  2  50 

*  Bartlett's  Mechanical  Drawing Svo,  3  00 

*  "                    "                  "       Abridged  Ed Svo,  150 

*  Bartlett  and  Johnson's  Engineering  Descriptive  Geometry Svo,  1  50 

*  Burr's  Ancient  and  Modern  Engineering  and  the  Isthmian  Canal Svo,  3  50 

Carpenter's  Experimental  Engineering Svo,  6  00 

Heating  and  Ventilating  Buildings Svo,  4  00 

*  Clerk's  The  Gas,  Petrol  and  Oil  Engine Svo,  4  00 

Compton's  First  Lessons  in  Metal  Working 12mo,  1  50 

Compton  and  De  Groodt's  Speed  Lathe 12mo,  1  50 

Coolidge's  Manual  of  Drawing Svo,  paper,  1  00 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  En- 
gineers  Oblong  4to,  2  50 

Cromwell's  Treatise  on  Belts  and  Pulleys 12mo,  1  50 

Treatise  on  Toothed  Gearing 12mo,  1  50 

Dingey's  Machinery  Pattern  Making 12mo,  2  00 

Durley's  Kinematics  of  Machines Svo,  4  00 

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Flather's  Dynamometers  and  the  Measurement  of  Power 12mo,  3  00 

Rope  Driving 12mo,  2  00 

Gill's  Gas  and  Fuel  Analysis  for  Engineers 12mo,  1  25 

Goss's  Locomotive  Sparks Svo,  2  00 

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*  Hobart  and  Ellis's  High  Speed  Dynamo  Electric  Machinery Svo,  6  00 

Hutton's  Gas  Engine Svo,  5  00 

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Machine  Design:  _ 

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Kerr's  Power  and  Power  Transmission Svo,  2  00 

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Leonard's  Machine  Shop  Tools  and  Methods Svo,  4  00 

*  Levin's  Gas  Engine Svo,  4  00 

*  Lorenz's  Modern  Refrigerating  Machinery.   (Pope,  Haven,  and  Dean).  .Svo,  4  00 
MacCord's  Kinematics;  or.  Practical  Mechanism Svo,  5  00 

Mechanical  Drawing 4to,  4  00 

Velocity  Diagrams .Svo,  1  50 

MacFarland's  Standard  Reduction  Factors  for  Gases Svo,  1  50 

Mahan's  Industrial  Drawing.     (Thompson.) Svo,  3  50 

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Oberg's  Handbook  of  Small  Tools Large  12mo. 

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larged  8vo, 

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*  Reid's  Mechanical  Drawing.      (Elementary  and  Advanced.) 8vo, 

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Richards's  Compressed  Air 12mo. 

Robinson's  Principles  of  Mechanism 8vo, 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo, 

Smith  (A.  W.)  and  Marx's  Machine  Design 8vo, 

Smith's  (O.)  Press-working  of  Metals 8vo, 

Sorel's  Carbureting  and  Combustion  in  Alcohol  Engines.     (Woodward  and 

Preston.) Large  12mo, 

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Thurston's  Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics. 

12mo, 

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Wood's  Turbines 8vo, 


MATERIALS   OF  ENGINEERING. 

*  Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering 8vo,  7  50 

Church's  Mechanics  of  Engineering 8vo,  6  00 

*  Greene's  Structural  Mechanics 8vo,  2  60 

*  HoUey's  Lead  and  Zinc  Pigments Large  12mo  3  00 

Holley  and  Ladd's  Analysis  of  Mixed  Paints,  Color  Pigments,  and  Varnishes. 

Large  12mo,  2  50 
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Steels,  Steel- Making  Alloys  and  Graphite Large  12mo,  3  00 

Johnson's  (J.  B.)  Materials  of  Construction 8vo,  6  00 

Keep's  Cast  Iron 8vo,  2  50 

Lanza's  Applied  Mechanics 8vo,  7  50 

Lowe's  Paints  for  Steel  Structures 12mo,  1  00 

Maire's  Modem  Pigments  and  theit  Vehicles 12mo,  2  00 

Maurer's  Technical  Mechanics 8vo,  4  GO 

Merriman's  Mechanics  of  Materials 8vo,  5  00 

*  Strength  of  Materials 12mo,  1  00 

Metcalf's  Steel.     A  Manual  for  Steel-users 12mo,  2  00 

Sabin's  Industrial  and  Artistic  Technology  of  Paint  and  Varnish 8vo,  3  00 

Smith's  ((A.  W.)  Materials  of  Machines 12mo,  1  00 

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Part  II.     Iron  and  Steel 8vo,  3  50 

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Constituents 8vo,  2  50 

Wood's  (De  V.)  Elements  of  Analytical  Mechanics 8vo,  3  00 

Treatise  on    the    Resistance   of    Materials   and    an   Appendix   on    the 

Preservation  of  Timber 8vo,  2  00 

Wood's  (M.  P.)  Rustless  Coatings:    Corrosion  and  Electrolysis  of  Iron  and 

Steel 8vo,  4  OO 

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STEAM-ENGINES    AND    BOILERS. 

Berry's  Temperature-entropy  Diagram 12mo, 

Camot's  Reflections  on  the  Motive  Power  of  Heat.      (Thurston.) 12mo, 

Chase's  Art  of  Pattern  Making 12mo, 

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Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  ..  .  l6mo,  mor. 

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Goss's  Locomotive  Performance 8vo, 

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Hutton's  Heat  and  Heat-engines Svo, 

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Kent's  Steam  Boiler  Economy Svo, 

Kneass's  Practice  and  Theory  of  the  Injector Svo, 

MacCord's  Slide-valves Svo, 

Meyer's  Modern  Locomotive  Construction 4to, 

Moyer's  Steam  Turbine Svo, 

Peabody's  Manual  of  the  Steam-engine  Indicator 12mo, 

Tables  of  the  Properties  of  Steam  and  Other  Vapors  and  Temperature- 
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Thermodynamics  of  the  Steam-engine  and  Other  Heat-engines.  .  .  .  Svo, 

Valve-gears  for  Steam-engines Svo, 

Peabody  and  Miller's  Steam-boilers Svo, 

Pupin's  Thermodynamics  of  Reversible  Cycles  in  Gases  and  Saturated  Vapors. 

(Osterberg.) , 12mo, 

Reagan's  Locomotives:  Simple,  Compound,  and  Electric.     New  Edition. 

Large  I2mo, 

Sinclair's  Locomotive  Engine  Running  and  Management 12mo, 

Smart's  Handbook  of  Engineering  Laboratory  Practice 12mo, 

Snow's  Steam-boiler  Practice Svo, 

Spangler's  Notes  on  Thermodynamics 12mo, 

Valve-gears Svo, 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering Svo, 

Thomas's  Steam-turbines Svo, 

Thurston's  Handbook  of  Engine  and  Boiler  Trials,  and  the  Use  of  the  Indi- 
cator and  the  Prony  Brake Svo, 

Handy  Tables Svo, 

Manual  of  Steam-boilers,  their  Designs,  Construction,  and  Operation  Svo, 

Manual  of  the  Steam-engine 2  vols.,  Svo, 

Part  I.     History,  Structure,  and  Theory Svo, 

Part  II.     Design,  Construction,  and  Operation Svo, 

Wehrenfennig's  Analysis  and  Softening  of  Boiler  Feed-water.    (Patterson.) 

Svo, 

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Whitham's  Steam-engine  Design Svo, 

Wood's  Thermodynamics,  Heat  Motors,  and  Refrigerating  Machines.  .  .Svo, 


MECHANICS    PURE   AND    APPLIED. 

Church's  Mechanics  of  Engineering Svo,  6  00 

*  Mechanics  of  Internal  Works Svo,  1  50 

Notes  and  Examples  in  Mechanics Svo,  2  00 

Dana's  Text-book  of  Elementary  Mechanics  for  Colleges  and  Schools  .12mo,  1  50 
Du  Bois's  Elementary  Principles  of  Mechanics: 

Vol.    I.     Kinematics Svo,  3  50 

Vol.  II.     Statics Svo.  4  GO 

Mechanics  of  Engineering.     Vol.    I Small  4to,  7  50 

Vol.  II Small  4to,  10  00 

*  Greene's  Structural  Mechanics Svo,  2  50 

*  Hartmann's  Elementary  Mechanics  for  Engineering  Students 12mo,  1  25 

James's  Kinematics  of  a  Point  and  the  Rational  Mechanics  of  a  Particle. 

Large  12mo.  2  00 

*  Johnson's  (W.  W.)  Theoretical  Mechanics 12mo,  3  00 

Lanza's  Applied  Mechanics Svo,  7  50 

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•  Martin's  Text  Book  on  Mechanics.  Vol.  I,  Sutics 12mo, 

*  Vol.  II,  Kinematics  and  Kinetics.  12mo, 
Maurer's  Technical  Mechanics 8vo. 

♦  Merriman's  Elements  of  Mechanics 12mo, 

Mechanics  of  Materials 8vo, 

♦  Michie's  Elements  of  Analytical  Mechanics 8vo. 

Robinson's  Principles  of  Mechanism 8vo, 

Sanborn's  Mechanics  Problems Large  12mo, 

Schwamb  and  Merrill's  Elemen.'^s  of  Mechanism 8vo, 

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MEDICAL. 

•  Abderhalden's  Physiological  Chemistry  in  Thirty  Lectures.     (Hall  and 

Defren.) 8vo.  5  00 

von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.) 12mo,  1  00 

Bolduan's  Immune  Sera 12mo,  1  50 

Bordet's  Studies  in  Immunity.     (Gay.) 8vo,  6  00 

*  Chapin's  The  Sources  and  Modes  of  Infection Large  12mo,  3  00 

Davenport's  Statistical  Methods  with  Special  Reference  to  Biological  Varia- 
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de  Fursac's  Manual  of  Psychiatry.     (Rosanoflf  and  Collins.).. .  .Large  12mo,  2  50 

Hammarsten's  Text-book  on  Physiological  Chemistry.      (Mandel.) Svo,  4  00 

Jackson's  Directions  for  Laboratory  Work  in  Physiological  Chemistry.  .Svo,  1  25 

Lassar-Cohn's  Praxis  of  Urinary  Analysis.     (Lorenz.) 12mo,  1  00 

Mandel's  Hand-book  for  the  Bio-Chemical  Laboratory 12mo.  1  50 

*  Nelson's  Analysis  of  Drugs  and  Medicines l2mo,  3  00 

♦  Pauli's  Physical  Chemistry  in  the  Service  of  Medicine.      (Fischer. )..12mo,  1  25 

*  Pozzi-Escot's  Toxins  and  Venoms  and  their  Antibodies.     (Cohn.).  .  12mo,  1  00 

Rostoski's  Serum  Diagnosis.      (Bolduan.) 12mo,  1  00 

Ruddiman's  Incompatibilities  in  Prescriptions Svo,  2  00 

Whys  in  Pharmacy 12mo.  1  00 

Salkowski's  Physiological  and  Pathological  Chemistry,     (Omdorff.) Svo,  2  50 

*  Satterlee's  Outlines  of  Human  Embryology 12mo,  1  25 

Smith's  Lecture  Notes  on  Chemistry  for  Dental  Students Svo,  2  50 

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METALLURGY. 

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BoUand's  Encyclopedia  of  Founding  and  Dictionary  of  Foundry  Terms  used 

in  the  Practice  of  Moulding 12mo, 

Iron  Foundec 12mo, 

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Borchers's  Metallurgy.     (Hall  and  Hayward.)     (In  Press.) 

Douglas's  Untechnical  Addresses  on  Technical  Subjects 12mo, 

Goesel's  Minerals  and  Metals:  A  Reference  Book 16mo,  mor. 

*  Iles's  Lead-smelting 12mo, 

Johnson's    Rapid    Methods  for   the  Chemical   Analysis  of  Special   Steels, 

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Keep's  Cast  Iron Svo, 

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Metcalf's  Steel.     A  Manual  for  Steel-users 12mo, 

Minet's  Production  of  Aluminum  and  its  Industrial  Use.      (Waldo.).  .  12mo, 

•  Ruer's  Elements  of  Metallography.     (Mathewson.) Svo, 

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Smith's  Materials  of  Machines 12mo,  $1  00 

Tate  and  Stone's  Foundry  Practice 12mo,  2  00 

Thurston's  Materials  of  Engineering.     In  Three  Parts 8vo,  8  00 

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Part  II.    Iron  and  Steel 8vo,  3  50 

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Constituents 8vo,  2  50 

Ulke's  Modern  Electrolytic  Copper  Refining 8vo,  3  00 

West's  American  Foundry  Practice 12mo,  2  50 

Moulders'  Text  Book 12mo.  2  50 


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Baskerville's  Chemical  Elements.      (In  Preparation.) 

*  Browning's  Introduction  to  the  Rarer  Elements Svo, 

Brush's  Manual  of  Determinative  Mineralogy.      (Penfield.) Svo, 

V  Butler's  Pocket  Hand-book  of  Minerals 16mo,  mor. 

Chester's  Catalogue  of  Minerals Svo,  paper. 

Cloth, 

*  Crane's  Gold  and  Silver Svo, 

Dana's  First  Appendix  to  Dana's  New  "System  of  Mineralogy".  .Large  Svo, 
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Large  Svo, 

Manual  of  Mineralogy  and  Petrography 12mo, 

Minerals  and  How  to  Study  Them 12mo, 

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Text-book  of  Mineralogy Svo, 

Douglas's  Untechnical  Addresses  on  Technical  Subjects 12mo, 

Eakle's  Mineral  Tables Svo, 

Eckel's  Stone  and  Clay  Products  Used  in  Engineering.      (In  Preparation.) 

Goesel's  Minerals  and  Metals:  A  Reference  Book 16mo,  mor. 

Groth's  The  Optical  Properties  of  Crystals.     (Jackson.)      (In  Press.) 
Groth's  Introduction  to  Chemical  Crystallography  (Marshall) 12mo, 

*  Hayes's  Handbook  for  Field  Geologists 16mo,  mor. 

Iddings's  Igneous  Rocks Svo, 

Rock  Minerals Svo, 

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pipe  12mo, 

Merrill's  Non-metallic  Minerals:  Their  Occurrence  and  Uses Svo, 

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*  Ries's  Clays:  Their  Occurrence,  Properties  and  Uses Svo, 

*  Ries  and  Leighton's  History  of  the  Clay-working  Industry  of  the  United 

States Svo, 

Rowe's  Practical  Mineralogy  Simplified.      (In  Press.) 

*  Tillman's  Text-book  of  Important  Minerals  and  Rocks Svo, 

Washington's  Manual  of  the  Chemical  Analysis  of  Rocks Svo, 


MINING. 

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*  Crane's  Gold  and  Silver Svo,  5  00 

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*  Svo.  mor.  5  00 

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Douglas's  Untechnical  Addresses  on  Technical  Subjects 12mo,  1  00 

Eissler's  Modem  High  Explosives Svo,  4  00 

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Goesel's  Minerals  and  Metals:  A  Reference  Book 16mo,  mor.  $3  00 

Vlhlseng's  Manual  of  Mining 8vo,  5  00 

♦  Iles's  Lead  Smelting 12mo.  2  00 

Peele's  Compressed  Air  Plant  for  Mines 8vo,  3  00 

Riemer's  Shaft  Sinking  Under  Difficult  Conditions.     (Coming  and  Peele.)8vo,  3  00 

♦  Weaver's  Military  Explosives 8vo,  3  00 

Wilson's  Hydraulic  and  Placer  Mining.     21  edition,  rewritten 12mo,  2  60 

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SANITARY   SCIENCE. 

Association  of  State  and  National  Food  and  Dairy  Departments,  Hartford 

Meeting,  1906 8vo, 

Jamestown  Meeting,  1907 ' 8vo, 

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*  Chapin's  The  Sources  and  Modes  of  Infection Large  12mo, 

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Tumeaure  and  Russell's  Public  Water-supplies Svo, 

Venable's  Garbage  Crematories  in  America Svo, 

Method  and  Devices  for  Bacterial  Treatment  of  Sewage Svo, 

Ward  and  Whipple's  Freshwater  Biology.     (In  Press.) 

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*  Typhoid  Fever Large  12mo, 

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18 


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MISCELLANEOUS. 

Emmons's  Geological  Guide-book  of  the  Rocky  Mountain  Excursion  of  the 

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